Engineering 
Library 


THEORY  AND  CALCULATION 

OF 

ALTERNATING-CURRENT  PHENOMENA 


McGraw-Hill  BookCompany 


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Metallurgical  and  Chemical  Engineering  Power 


THEORY  AND  CALCULATION 

OF 

ALTERNATING  CURRENT 
PHENOMENA 


BY 

CHARLES  PROTEUS  STEINMETZ,  A.M.,  PH.D. 

\\ 


FIFTH  EDITION 
THOROUGHLY  REVISED  AND  ENTIRELY  RESET 


McGRAW-HILL  BOOK  COMPANY,  INC. 

239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 

6  &  8  BOUVERIE  ST.,  E.  C. 

1916 


-SP- 

Engineering 
Library 


COPYRIGHT,  1916,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 

COPYRIGHT,  1908,  BY  THE 
McGRAW  PUBLISHING  COMPANY 

COPYRIGHT,  1900,  BY 
ELECTRICAL  WORLD  AND  ENGINEER  (INCORPORATED) 


..,-,    *,-,.•'_     *  >     •     , 

?'U&;..; 


THE . MAPLK. PRESS- YORK. PA 


De&fcateD 

TO   THE 
MEMORY  OF  MY  FATHER 

CARL  HEINRICH  STEINMETZ 


338054 


PREFACE  TO  FIFTH  EDITION 

When  the  first  edition  of  " Alternating-Current  Phenomena" 
appeared  nearly  twenty  years  ago,  it  was  a  small  volume  of  424 
pages.  From  this,  it  grew  to  a  volume  of  746  pages  in  the  fourth 
edition,  which  appeared  eight  years  ago.  Since  that  time,  the 
advance  of  electrical  engineering  has  been  more  rapid  than  ever 
before,  and  any  attempt  to  treat  adequately  in  one  volume  all 
the  new  material  developed  in  the  last  eight  years,  and  the 
material  which  during  this  time  has  become  of  such  importance 
as  to  require  more  extensive  consideration,  thus  became  out  of  the 
question.  It  was  found  necessary,  therefore,  to  divide  the  work 
into  three  volumes.  In  the  following,  under  the  old  title  "  Theory 
and  Calculation  of  Alternating-Current  Phenomena,"  is  included 
only  the  discussion  of  the  most  common  and  general  phenomena 
and  apparatus,  old  and  new,  revised  and  expanded  so  as  to  bring 
it  up  to  our  present  knowledge.  All  the  material,  some  partly 
old,  but  mostly  new,  which  could  not  find  place  in  the  present 
volume,  will  be  presented  in  two  supplementary  volumes,  under 
the  titles  "  Theory  and  Calculation  of  Electric  Circuits,"  and 
" Theory  and  Calculation  of  Electrical  Apparatus." 

In  the  study  of  electrical  engineering  theory,  it  is  recommended 
to  read  first  Part  I  of  "  Theoretical  Elements  of  Electrical  Engi- 
neering," and  then  the  first  three  sections  of  "Alternating  Cur- 
rent Phenomena,"  but  to  parallel  the  reading  with  that  of  the 
chapters  of  " Engineering  Mathematics,"  which  deal  with  the 
mathematics  involved.  Then  Sections  IV  to  VII  of  "Alter- 
nating-Current Phenomena"  should  be  studied  simultaneously 
with  the  corresponding  discussion  of  the  apparatus  in  the  Part 
II  of  "Theoretical  Elements."  Following  this  should  be  taken 
up  the  study  of  "Theory  and  Calculation  of  Electric  Circuits/' 
"Theory  and  Calculation  of  Electrical  Apparatus,"  and  the  first 
three  sections  of  "Transient  Phenomena,"  and,  finally,  the  study 
of  "Electric  Discharges,  Waves  and  Impulses"  and  of  the  fourth 
section  of  "Transient  Phenomena." 

In  the  present  edition  of  "Alternating-Current  Phenomena," 
the  crank  diagram  of  vector  representation,  and  the  symbolic 
method  based  on  it,  which  denotes  the  inductive  reactance  by 

vii 


viii  PREFACE  TO  FIFTH  EDITION 

Z  =  r  +  jx}  have  been  adopted  in  conformity  with  the  decision 
of  the  International  Electrical  Congress  of  Turin,  but  the  time 
diagram  or  polar  coordinate  system  has  been  explained  and  dis- 
cussed in  Chapter  VII,  since  the  crank  diagram  is  somewhat 
inferior  to  the  polar  diagram,  as  it  is  limited  to  sine  waves,  and 
the  time  diagram  will  thus  remain  in  use  when  dealing  with 
general  waves  and  their  graphic  reduction. 

CHARLES  P.  STEINMETZ. 

SCHENECTADY,    N.  Y., 

May,  1916. 


PREFACE  TO  FIRST  EDITION 

THE  following  volume  is  intended  as  an  exposition  of  the 
methods  which  I  have  found  useful  in  the  theoretical  investiga- 
tion and  calculation  of  the  manifold  phenomena  taking  place  in 
alternating-current  circuits,  and  of  their  application  to  alternat- 
ing-current apparatus. 

While  the  book  is  not  intended  as  first  instruction  for  a  begin- 
ner, but  presupposes  some  knowledge  of  electrical  engineering,  I 
have  endeavored  to  make  it  as  elementary  as  possible,  and  have 
therefore  used  only  common  algebra  and  trigonometry,  practi- 
cally excluding  calculus,  except  in  §§144  to  151  and  Appendix  II; 
and  even  §§144  to  151  have  been  paralleled  by  the  elementary 
approximation  of  the  same  phenomenon  in  §§140  to  143. 

All  the  methods  used  in  the  book  have  been  introduced  and 
explicitly  discussed,  with  examples  of  their  application,  the  first 
part  of  the  book  being  devoted  to  this.  In  the  investigation  of 
alternating-current  phenomena  and  apparatus,  one  method  only 
has  usually  been  employed,  though  the  other  available  methods 
are  sufficiently  explained  to  show  their  application. 

A  considerable  part  of  the  book  is  necessarily  devoted  to  the 
application  of  complex  imaginary  quantities,  as  the  method 
which  I  found  most  useful  in  dealing  with  alternating-current 
phenomena;  and  in  this  regard  the  book  may  be  considered  as 
an  expansion  and  extension  of  my  paper  on  the  application  of 
complex  imaginary  quantities  to  electrical  engineering,  read  be- 
fore the  International  Electrical  Congress  at  Chicago,  1893.  The 
complex  imaginary  quantity  is  gradually  introduced,  with  full 
explanations,  the  algebraic  operations  with  complex  quantities 
being  discussed  in  Appendix  I,  so  as  not  to  require  from  the  reader 
any  previous  knowledge  of  the  algebra  of  the  complex  imaginary 
plane. 

While  those  phenomena  which  are  characteristic  of  polyphase 
systems,  as  the  resultant  action  of  the  phases,  the  effects  of  un- 
balancing, the  transformation  of  polyphase  systems,  etc.,  have 
been  discussed  separately  in  the  last  chapters,  many  of  the  in- 
vestigations in  the  previous  parts  of  the  book  apply  to  poly- 
phase systems  as  well  as  single-phase  circuits,  as  the  chapters  on 
induction  motors,  generators,  synchronous  motors,  etc. 

ix 


x  PREFACE  TO  FIRST  EDITION 

A  part  of  the  book  is  original  investigation,  either  published 
here  for  the  first  time,  or  collected  from  previous  publications 
and  more  fully  explained.  Other  parts  have  been  published  be- 
fore by  other  investigators,  either  in  the  same,  or  more  frequently 
in  a  different  form. 

I  have,  however,  omitted  altogether  literary  references,  for  the 
reason  that  incomplete  references  would  be  worse  than  none, 
while  complete  references  would  entail  the  expenditure  of  much 
more  time  than  is  at  my  disposal,  without  offering  sufficient  com- 
pensation; since  I  believe  that  the  reader  who  wants  informa- 
tion on  some  phenomenon  or  apparatus  is  more  interested  in 
the  information  than  in  knowinjg  who  first  investigated  the 
phenomenon. 

Special  attention  has  been  given  to  supply  a  complete  and  ex- 
tensive index  for  easy  reference,  and  to  render  the  book  as  free 
from  errors  as  possible.  Nevertheless,  it  probably  contains  some 
errors,  typographical  and  otherwise;  and  I  will  be  obliged  to  any 
reader  who  on  discovering  an  error  or  an  apparent  error  will 
notify  me. 

I  take  pleasure  here  in  expressing  my  thanks  to  Messrs.  W.  D. 
WEAVER,  A.  E.  KENNELLY,  and  TOWNSEND  WOLCOTT,  for  the 
interest  they  have  taken  in  the  book  while  in  the  course  of  pub- 
lication, as  well  as  for  the  valuable  assistance  given  by  them  in 
correcting  and  standardizing  the  notation  to  conform  to  the 
international  system,  and  numerous  valuable  suggestions  regard- 
ing desirable  improvements. 

Thanks  are  due  to  the  publishers,  who  have  spared  no 
effort  or  expense  to  make  the  book  as  creditable  as  possible 
mechanically. 

CHARLES  PROTEUS  STEINMETZ. 

January,  1897. 


CONTENTS 

SECTION  I 

METHODS  AND  CONSTANTS 
CHAPTER  I.     INTRODUCTION. 

PAGE 

1.  Fundamental  laws  of  continuous-current  circuits.  1 

2.  Impedance,  reactance,  effective  resistance.  2 

3.  Electromagnetism  as  source  of  reactance.  2 

4.  Capacity  as  source  of  reactance.  4 

5.  Joule's  law  and  power  equation  of  alternating  circuit.  5 

6.  Fundamental  wave  and  higher  harmonics,  alternating  waves 

with  and  without  even  harmonics.  5 

7.  Alternating  waves  as  sine  waves.  8 

8.  Experimental  determination  and  calculation  of  reactances.  8 

CHAPTER  II.     INSTANTANEOUS  VALUES  AND  INTEGRAL  VALUES. 

9.  Integral  values  of  wave.  1 1 

10.  Ratio  of  mean  to  maximum  to  effective  value  of  wave.  13 

11.  General  alternating-current  wave.  14 

12.  Measurement  of  values.  15 

CHAPTER  III.    LAW  OP  ELECTROMAGNETIC  INDUCTION. 

13.  Induced  e.m.f.  mean  value.  16 

14.  Induced  e.m.f.  effective  value.  17 

15.  Inductance  and  reactance.  17 

CHAPTER  IV.     VECTOR  REPRESENTATION. 

16.  Crank  diagram  of  sine  wave.  19 

17.  Representation  of  lag  and  lead.  20 

18.  Parallelogram   of   sine  waves,   Kirchhoff's  laws,   and  energy 

equation.  21 

19.  Non-inductive  circuit  fed  over  inductive  line,  example.  22 

20.  Counter  e.m.f.  and  component  of  impressed  e.m.f.  23 

21.  Example,  continued.  24 

22.  Inductive  circuit  and  circuit  with  leading  current  fed  over  in- 

ductive line.     Alternating-current  generator.  25 

23.  Vector  diagram  of  alternating-current  transformer,  example.  26 

24.  Continued.  28 

CHAPTER  V.     SYMBOLIC  METHOD. 

25.  Disadvantage  of  graphic  method  for  numerical  calculation.  30 

26.  Trigonometric  calculation.  31 ' 

xi 


xii  CONTENTS 

PAGE 

27.  Rectangular  components  of  vectors.  31 

28.  Introduction  of  j  as  distinguishing  index.  32 

29.  Rotation  of  vector  by  180°  and  90°.    j  =  \/~^~l.  32 

30.  Combination  of  sine  waves  in  symbolic  expression.  33 

31.  Resistance,  reactance,  impedance,  in  symbolic  expression.  34 

32.  Capacity  reactance  in  symbolic  representation.  35 

33.  Kirchhoff's  laws  in  symbolic  representation.  36 

34.  Circuit  supplied  over  inductive  line,  example.  37 

35.  Products  and  ratios  of  complex  quantities.  37 

CHAPTER  VI.     TOPOGRAPHIC  METHOD. 

36.  Ambiguity  of  vectors.  39 

37.  Instance  of  a  three-phase  system.  39 

38.  Three-phase  generator  on  balanced  load.  41 

39.  Cable  with  distributed  capacity  and  resistance.  42 

40.  Transmission  line  with  inductance,  capacity,  resistance,  and 

leakage.  43 

41.  Line  characteristic  at  90°  lag.  45 

CHAPTER  VII.     POLAR  COORDINATES  AND  POLAR  DIAGRAM. 

42.  Polar  Coordinates  46 

43.  Sine  wave,  vector  representation  or  time  diagram.  46 

44.  Parallelogram   of  sine  waves,    Kirchhoff's  Laws  and   power 

equation.  48 

45.  Comparison  of  time  diagram  and  crank  diagram.  49 

46.  Comparison  of  corresponding  symbolic  methods.  51 

47.  Disadvantage  of    crank    diagram.     General    wave  end    its 

equivalent  sine  wave  in  time  diagram.  52 


SECTION  II 
CIRCUITS 
CHAPTER  VIII.     ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE. 

48.  Combination  of  resistances  and  conductances  in  series  and 

in  parallel.  54 

49.  Combination     of     impedances.     Admittance,     conductance, 

susceptance.  55 

50.  Relation  between  impedance,  resistance,  reactance,  and  ad- 

mittance, conductance,  susceptance.  56 

51.  Dependence  of  admittance,  conductance,  susceptance,  upon 

resistance  and  reactance.     Combination  of  impedances  and 
admittances.  57 

52.  Measurements  of  admittance  and  impedance.  59 


CONTENTS  xiii 

CHAPTER  IX.     CIRCUITS  CONTAINING  RESISTANCE,  INDUCTANCE,  AND 
CAPACITY. 

PAGE 

53.  Introduction.  60 

54.  Resistance  in  series  with  circuit.  60 

55.  Reactance  in  series  with  circuit.  63 

56.  Discussion  of  examples.  65 

57.  Reactance  in  series  with  circuit.  67 

58.  Impedance  in  series  with  circuit.  69 

59.  Continued.  70 

60.  Example.  70 

61.  Compensation  for  lagging  currents  by  shunted  condensance.  72 

62.  Complete  balance  by  variation  of  shunted  condensance.  73 

63.  Partial  balance  by  constant  shunted  condensance.  75 

64.  Constant  potential — constant-current  transformation.  76 

CHAPTER  X.     RESISTANCE  AND  REACTANCE  OF  TRANSMISSION  LINES. 

65.  Introduction.  78 

66.  Non-inductive  receiver  circuit  supplied  over  inductive  line.  79 

67.  Example.  81 

68.  Maximum  power  supplied  over  inductive  line.  82 

69.  Dependence  of  output  upon  the  susceptance  of  the  receiver 

circuit.  82 

70.  Dependence  of  output  upon  the  conductance  of  the  receiver 

circuit.  84 

71.  Summary.  85 

72.  Example.  86 

73.  Condition  of  maximum  efficiency.  88 

74.  Control  of  receiver  voltage  by  shunted  susceptance.  89 

75.  Compensation  for  line  drop  by  shunted  susceptance.  90 

76.  Maximum  output  and  discussion.  91 

77.  Example.  92 

78.  Maximum  rise  of  potential  in  receiver  circuit.  94 

79.  Summary  and  examples.  96 

CHAPTER  XI.     PHASE  CONTROL. 

80.  Effect  of  the  current  phase  in  series  reactance,  on  the  voltage.  97 

81.  Production  of  reactive  currents  by  variation  of  field  of  syn- 

chronous machines.  98 

82.  Fundamental  equations  of  phase  control.  99 

83.  Phase  control  for  unity  power-factor  supply.  100 

84.  Phase  control  for  constant  receiver-voltage.  102 

85.  Relations  between  supply  voltage,  no-load  current,  full-load 

current  and  maximum  output  current.  104 

86.  Phase  control  by  series  field  of  converter.  105 

87.  Multiple-phase  control  for  constant  voltage.  107 

88.  Adjustment  of  converter  field  for  phase  control.  109 


xiv  CONTENTS 

SECTION  III 
POWER  AND  EFFECTIVE  CONSTANTS 

CHAPTER  XII.     EFFECTIVE  RESISTANCE  AND  REACTANCE. 

PAGE 

89.  Effective  resistance,  reactance,  conductance,  and  susceptance.  Ill 

90.  Sources  of  energy  losses  in  alternating-current  circuits.  112 

91.  Magnetic  hystersis.  113 

92.  Hysteretic  cycles  and  corresponding  current  waves.  114 

93.  Wave-shape  distortion  not  due  to  hysteresis.  117 

94.  Action    of    air-gap    and    of    induced    current    on    hysteretic 

distortion.  119 

95.  Equivalent  sine  wave  and  wattless  higher  harmonics.  120 

96.  True  and  apparent  magnetic  characteristic.  121 

97.  Angle  of  hysteretic  advance  of  phase.  122 

98.  Loss  of  energy  by  molecular  magnetic  friction.  123 

99.  Effective  conductance,  due  to  magnetic  hysteresis.  126 

100.  Absolute  admittance  of  iron-clad  circuits  and  angle  of  hysteretic 

advance.  129 

101.  Magnetic  circuit  containing  air-gap.  131 

102.  Electric  constants  of  circuit  containing  iron.  132 

103.  Conclusion.  133 

104.  Effective  conductance  of  eddy  currents.  135 

CHAPTER  XIII.     FOUCAULT  OR  EDDY  CURRENTS. 

105.  Advance  angle  of  eddy  currents.  136 

106.  Loss  of  power  by  eddy  currents,  and  coefficient  of  eddy  currents.  137 

107.  Laminated  iron.  138 

108.  Iron  wire.  140 

109.  Comparison  of  sheet  iron  and  iron  wire.  141 

110.  Demagnetizing  or  screening  effect  of  eddy  currents.  142 

111.  Continued.  143 

112.  Large  eddy  currents.  144 

113.  Eddy  currents  in  conductor  and  unequal  current  distribution.  144 

114.  Continued.  -145 

115.  Mutual  inductance.  147 

CHAPTER  XIV.     DIELECTRIC  LOSSES. 

116.  Dielectric  hysteresis.  150 

117.  Leakage.     Dynamic    current    and    displacement    current. 

Power  factor.  151 

118.  Effect  of  frequency.     Heterogeneous  dielectric.  153 

119.  Power  factor  and  stress  in  compound  dielectric.  154 

120.  Variation  of  power  factor  with  frequency.  156 

121.  Dielectric  circuit  and  dynamic  circuit.     Admittance  and  im- 

pedance.    Admittivity.  158 

122.  Study  of  dielectric  field.  160 

123.  Corona.     Dielectric  strength  and  gradient.  161 


CONTENTS  xv 

PAGE 

124.  Corona  and  disruption.     Numerical  instance.  161 

125.  Energy    distance,    disruptive    gradient    and    visual    corona 

gradient.  164 

126.  Law  of  corona  on  parallel  conductors.  165 

CHAPTER  XV.     DISTRIBUTED    CAPACITY,    INDUCTANCE,   RESISTANCE, 
AND  LEAKAGE. 

127.  Energy  components  and  wattless  components.  168 

128.  Distributed  capacity.  168 

129.  Magnitude  of  charging  current  of  transmission  lines.  170 

130.  Line  capacity  represented  by  one  condenser  shunted  across 

middle  of  line.  171 

131.  Distributed  capacity,  inductance,  conductance  and  resistance.  172 

132.  Constants  of  transmission  line.  174 

133.  Oscillating  functions  of  distance.     Approximate  calculation.  175 

134.  Equations  of  transmission  line.  177 

CHAPTER  XVI.     POWER,     AND    DOUBLE-FREQUENCY    QUANTITIES    IN 
GENERAL. 

135.  Double  frequency  of  power.  179 

136.  Symbolic  representation  of  power.  180 

137.  Extra-algebraic  features  thereof.  182 

138.  Polar  coordinates.  183 

139.  Combination  of  powers.  184 

140.  Torque  as  double-frequency  product.  185 


SECTION  IV 
INDUCTION  APPARATUS 

CHAPTER  XVII.     THE  ALTERNATING-CURRENT  TRANSFORMER. 

141.  General.  187 

142.  Mutual  inductance  and  self-inductance  of  transformer.  187 

143.  Magnetic  circuit  of  transformer.  188 

144.  Continued.  189 

145.  Polar  diagram  of  transformer.  190 

146.  Example.  192 

147.  Diagram  for  varying  load.  196 

148.  Example.  197 

149.  Symbolic  method,  equations.  197 

150.  Continued.  199 

151.  Apparent  impedance  of  transformer.     Transformer  equivalent 

to  divided  circuit.  201 

152.  Continued.  202 

153.  Experimental  determination  of  transformer  constants.  205 

154.  Calculation  of  transformer  constants.  207 


xvi  CONTENTS 

CHAPTER  XVIII.     POLYPHASE  INDUCTION  MOTOR. 

PAGE 

155.  Slip  and  secondary  frequency.  208 

156.  Equations  of  induction  motor.  209 

157.  Magnetic  flux,  admittance,  and  impedance.  210 

158.  E.m.f.  212 

159.  Graphic  representation.  214 

160.  Continued.  215 

161.  Torque  and  power.  216 

162.  Power  of  induction  motors.  217 

163.  Maximum  torque.  219 

164.  Continued.  221 

165.  Maximum  power.  222 

166.  Starting  torque.  223 

167.  Equations  of  torque.  227 

168.  Synchronism.  229 

169.  Near  synchronism.  229 

170.  Numerical  example  of  induction  motor.  230 

171.  Calculation  of  induction-motor  curves.  232 

172.  Numerical  example.  235 

CHAPTER  XIX.     INDUCTION  GENERATOR. 

173.  Induction  generator.  237 

174.  Power-factor  of  induction  generator.  237 

175.  Constant  speed  induction  generator.  239 

176.  Induction  generator  and  synchronous  motor.  242 

CHAPTER  XX.     SINGLE-PHASE  INDUCTION  MOTOR. 

177.  Single-phase  induction  motor.  245 

178.  Starting  devices  of  single-phase  motor.  246 

179.  Polyphase  motor  on  single-phase  circuit.  247 

180.  Condenser  in  tertiary  circuit.  249 

181.  Speed  curves  with  condenser.  250 

182.  Monocyclic  starting  device.  253 

183.  Resistance-reactance  starter.  256 

184.  Discussion.  257 

SECTION  V 

SYNCHRONOUS  MACHINES 
CHAPTER  XXI.     ALTERNATE-CURRENT  GENERATOR. 

185.  Magnetic  reaction  of  lag  and  lead.  259 

186.  Self-inductance  and  synchronous  reactance.  261 

187.  Equations  of  alternator.  263 

188.  Numerical  instance,  field  characteristic.  264 

189.  Dependence  of  terminal  voltage  on  phase  relation.  266 

190.  Constant  potential  regulation.  267 

191.  Constant  current  regulation,  maximum  output.  270 


CONTENTS  xvii 
CHAPTER  XXII.     ARMATURE  REACTIONS  OF  ALTERNATORS. 

PAGE 

192.  Similarity  and  difference  between  armature  reaction  and  self- 

induction.  272 

193.  Graphic  representation  of  armature  reaction  and  self-induction.  273 

194.  Symbolic  representation.  274 

195.  Discussion:  synchronous    reactance    and    norminal  induced 

e.m.f.  276 

196.  Variability,  and  quadrature  components  in  space,  of  armature 

reaction  and  self-induction.  278 

197.  Graphic   representation   of  variable   armature   reaction   and 

self-induction.  279 

198.  Symbolic  representation.  281 

199.  Continued.  284 

200.  Regulation  curve  of  alternator.  287 

201.  Example.  288 

202.  Discussion.  290 

CHAPTER  XXIII.     SYNCHRONIZING  ALTERNATORS. 

203.  Introduction.  292 

204.  Rigid  mechanical  connection.  292 

205.  Uniformity  of  speed.  292 

206.  Synchronizing.  293 

207.  Running  in  synchronism.  293 

208.  Series  operation  of  alternators.  294 

209.  Equations  of  synchronous-running  alternators,  synchronizing 

power.  294 

210.  Special  case  of  equal  alternators  at  equal  excitation.  297 

211.  Numerical  example.  300 

CHAPTER  XXIV.     SYNCHRONOUS  MOTOR. 

212.  Graphic  method.  301 

213.  Continued.  302 

214.  Example.  303 

215.  Constant  impressed  e.m.f.  and  constant  current.  306 

216.  Constant  impressed  and  counter  e.m.f.  307 

217.  Constant  impressed  e.m.f.  and  maximum  efficiency.  310 

218.  Constant  impressed  e.m.f.  and  constant  output.  311 

219.  Analytical     method.     Fundamental    equations     and    power 

characteristic.  315 

220.  Maximum  output.  318 

221.  No  load.  319 
.     222.  Minimum  current.  321 

223.  Maximum  displacement  of  phase.  323 

224.  Constant  counter  e.m.f.  323 

225.  Numerical  example.  324 

226.  Discussion  of  results.  326 

227.  Phase  characteristics  of  synchronous  motor.  328 


xviii  CONTENTS 

PAGE 

228.  Example.  331 

229.  Load  curves  of  synchronous  motor.  334 

230.  Variable  armature  reaction  and  self-induction.  338 

231.  Synchronous  condenser.  339 

SECTION  VI 
GENERAL  WAVES 
CHAPTER  XXV.     DISTORTION  OP  WAVE-SHAPE,  AND  ITS  CAUSES. 

232.  Equivalent  sine  wave.  341 

233.  Cause  of  distortion.  341 

234.  Lack  of  uniformity  and  pulsation  of  magnetic  field.  342 

235.  Continued.  345 

236.  Pulsation  of  reactance.  348 

237.  Pulsation  of  reactance  in  reaction  machine.  348 

238.  General  discussion.  350 

239.  Pulsation  of  resistance,  arc.  350 

240.  Example.  351 

241.  Distortion  of  wave-shape  by  arc.  353 

242.  Discussion.  353 

243.  Calculation  of  example.  354 

244.  Separation  of  overtones  from  distorted  wave.  357 

245.  Resolution  of  exciting-current  wave  of  transformer.  360 

246.  Distortion  of  e.m.f.  wave  with  sine  wave  of  current,  in  iron-clad 

circuit.  361 

247.  Existence    and    absence    of    third    harmonic    in    three-phase 

system.  363 

248.  Suppression  of  third  harmonics  in  transformers  on  three-phase 

system.  364 

249.  Wave-shape  distortion  in  Y-connected  transformers.  385 

250.  Disappearance  of  distortion  by  delta  connection,  etc.  367 

CHAPTER  XXVI.     EFFECTS  OF  HIGHER  HARMONICS. 

251.  Distortion  of  wave-shape  by  triple  and  quintuple  harmonics. 

Some  characteristic  wave-shapes.  369 

252.  Effect  of  self-induction  and  capacity  on  higher  harmonics.  372 

253.  Resonance  due  to  higher  harmonics  in  transmission  lines.  373 

254.  Power  of  complex  harmonic  waves.  375 

255.  Three-phase  generator.  375 

256.  Decrease  of  hysteresis  by  distortion  of  wave-shape.  377 

257.  Increase  of  hysteresis  by  distortion  of  wave-shape.  377 

258.  Eddy  currents  and  effect  of  distorted  waves  on  insulation.  377 

CHAPTER  XXVII.     SYMBOLIC  REPRESENTATION  OF  GENERAL  ALTER- 
NATING WAVE. 

259.  Symbolic  representation.  379 

260.  Effective  values.  381 


CONTENTS  xix 

PAGE 

261.  Power,  torque,  etc.     Circuit-factor.  381 

262.  Resistance,  inductance,  and  capacity  in  series.  384 

263.  Apparent  capacity  of  condenser.  386 

264.  Synchronous  motor.  389 

265.  Induction  motor.  392 

SECTION  VII 
POLYPHASE  SYSTEMS 
CHAPTER  XXVIII.     GENERAL  POLYPAASE  SYSTEMS. 

266.  Definition  of  systems,  symmetrical  and  unsymmetrical  systems.  396 

267.  Flow  of  energy.     Balanced  and  unbalanced  systems.     Inde- 

pendent and  interlinked  systems.     Star  connection  and  ring 
connection.  396 

268.  Classification  of  polyphase  systems.  398 

CHAPTER  XXIX.     SYMMETRICAL  POLYPHASE  SYSTEMS. 

269.  General  equations  of  symmetrical  systems.  399 

270.  Particular  systems.  400 

271.  Resultant  m.m.f.  of  symmetrical  system.  401 

272.  Particular  systems.  403 

CHAPTER  XXX.     BALANCED  AND  UNBALANCED  POLYPHASE  SYSTEMS. 

273.  Flow  of  energy  in  single-phase  system.  405 

274.  Flow  of  energy  in  polyphase  systems,  balance  factor  of  system.  406 

275.  Balance  factor.  406 

276.  Three-phase  system,  quarter-phase  system.  407 

277.  Inverted  three-phase  system.  408 

278.  Diagrams  of  flow  of  energy.  408 

279.  Monocyclic  and  polycyclic  systems.  409 

280.  Power  characteristic  of  alternating-current  system.  409 

281.  The  same  in  rectangular  coordinates.  409 

282.  Main  power  axes  of  alternating-current  system.  414 

CHAPTER  XXXI.     INTERLINKED  POLYPHASE  SYSTEMS. 

283.  Interlinked  and  independent  systems.  415 

284.  Star  connection  and  ring  connection.     Y-connection  and  delta 

connection.  415 

285.  Continued.  417 

286.  Star  potential   and   ring   potential.     Star   current   and   ring 

current.     Y-potential  and  Y-current,   delta  potential  and 
delta  current.  417 

287.  Equations  of  interlinked  polyphase  systems.  417 

288.  Continued.  419 


xx  CONTENTS 

CHAPTER  XXXII.     TRANSFORMATION  OF  POLYPHASE  SYSTEMS. 

PAGE 

289.  Constancy  of  balance  factor.  422 

290.  Equations  of  transformation  of  polyphase  systems.  422 

291.  Three-phase  quarter-phase  transformation.  423 

292.  Some  of  the  more  common  polyphase  transformations.  425 

293.  Transformation  with  change  of  balance  factor.  430 

CHAPTER  XXXIII.     COPPER  EFFICIENCY  OF  SYSTEMS. 

294.  General  discussion.  431 

295.  Comparison  on  the  basis  of  equality  of  minimum  difference  of 

potential.  433 

296.  Comparison  on  the  basis  of  equality  of  maximum  difference  of 

potential  between  conductors.  437 

297.  Continued.  439 

298.  Comparison  on  the  basis  of  equality  of  maximum  difference  of 

potential  between  conductors  and  ground.  440 

CHAPTER  XXXIV.     METERING  OF  POLYPHASE  CIRCUIT. 

299.  General  equations.  442 

300.  Continued.  443 

301.  Three-phase  metering.  445 

302.  Discussion.  446 

CHAPTER  XXXV.     BALANCED  SYMMETRICAL  POLYPHASE  SYSTEMS. 

303.  Resolution  of  polyphase  system  into  constituent  single-phase 

systems.  448 

304.  Instance  of  calculation  of  transmission  line.  449 

305.  Resultant  effects  of  all  phases.  452 

306.  Three-phase  and  single-phase  admittance.  454 

307.  Three-phase  and  single-phase  impedance.  456 

CHAPTER  XXXVI.     THREE-PHASE  SYSTEMS. 

308.  General  equations.  457 

309.  Special    cases:  balanced    system,    one    branch    loaded,    two 

branches  loaded.  460 

CHAPTER  XXXVII.     QUARTER-PHASE  SYSTEM. 

310.  General  equations.  462 

311.  Special  cases:  balanced  system,  one  branch  loaded.  463 

APPENDIX.     ALGEBRA  OF  COMPLEX  IMAGINARY  QUANTITIES 

312.  Introduction.  466 

313.  Numeration,  addition,  multiplication,  involution.  466 


CONTENTS  xxi 

PAGE 

314.  Subtraction,  negative  number.  467 

315.  Division,  fraction.  468 

316.  Evolution  and  logarithmation.  468 

317.  Imaginary  unit,  complex  imaginary  number.  468 

318.  Review.  469 

319.  Algebraic  operations  with  complex  quantities.  470 

320.  Continued.  471 

321.  Roots  of  the  unit.  .  '        472 

322.  Rotation.  472 

323.  Complex  imaginary  plane.  472 

INDEX.  475 


SECTION  I 
METHODS  AND  CONSTANTS 


CHAPTER  I 
INTRODUCTION 

1.  In  the  practical  applications  of  electrical  energy,  we  meet 
with  two  different  classes  of  phenomena,  due  respectively  to  the 
continuous  current  and  to  the  alternating  current. 

The  continuous-current  phenomena  have  been  brought  within 
the  realm  of  exact  analytical  calculation  by  a  few  fundamental 

laws: 

g 

1.  Ohm's  law:  i  —  -,  where  r,  the  resistance,  is  a  constant 

of  the  circuit. 

2.  Joule's  law:  P  =  i2r,  where  P  is  the  power,  or  the  rate  at 
which  energy  is  expended  by  the  current,  i,  in  the  resistance,  r. 

3.  The    power    equation:  P0  =  ei,    where   PQ    is    the    power 
expended  in  the  circuit  of  e.m.f.,  e,  and  current,  i. 

4.  Kirchhoff's  laws: 

(a)  The  sum  of  all  the  e.m.fs.  in  a  closed  circuit  =  0,  if  the 
e.m.f.  consumed  by  the  resistance,  ir,  is  also  considered  as  a 
counter  e.m.f.,   and  all  the  e.m.fs.   are  taken  in  their  proper 
direction. 

(b)  The  sum  of  all  the  currents  directed  toward  a  distributing 
point  =  0. 

In  alternating-current  circuits,  that  is,  in  circuits  in  which  the 
currents  rapidly  and  periodically  change  their  direction,  these 
laws  cease  to  hold.  Energy  is  expended,  not  only  in  the  con- 
ductor through  its  ohmic  resistance,  but  also  outside  of  it;  energy 
is  stored  up  and  returned,  so  that  large  currents  may  exist 
simultaneously  with  high  e.m.fs.,  without  representing  any 
considerable  amount  of  expended  energy,  but  merely  a  surging 
t'o  and  fro  of  energy;  the  ohmic  resistance  ceases  to  be  the  deter- 

1 


ALT  I'll  \  ATING-CURRENT  PHENOMENA 

mining  factor  of  current  value;  currents  may  divide  into  com- 
ponents, each  of  which  is  larger  than  the  undivided  current,  etc. 

2.  In    place   of   the   above-mentioned   fundamental   laws   of 
continuous  currents,  we  find  in  alternating-current  circuits  the 

following : 

Q 
Ohm's  law  assumes   the  form   i  =  -,  where  z,  the  apparent 

resistance,  or  impedance,  is  no  longer  a  constant  of  the  circuit, 
but  depends  upon  the  frequency  of  the  currents;  and  in  circuits 
containing  iron,  etc.,  also  upon  the  e.m.f. 

Impedance,  z,  is,  in  the  system  of  absolute  units,  of  the  same 
dimension  as  resistance  (that  is,  of  the  dimension  It"1  =  velocity), 
and  is  expressed  in  ohms. 

It  consists  of  two  components,  the  resistance,  r,  and  the 
reactance,  x,  or 

z  =  Vr2  +  x2. 

The  resistance,  r,  in  circuits  where  energy  is  expended  only 
in  heating  the  conductor,  is  the  same  as  the  ohmic  resistance  of 
continuous-current  circuits.  In  circuits,  however,  where  energy 
is  also  expended  outside  of  the  conductor  by  magnetic  hysteresis, 
mutual  inductance,  dielectric  hysteresis,  etc.,  r  is  larger  than  the 
true  ohmic  resistance  of  the  conductor,  since  it  refers  to  the  total 
expenditure  of  energy.  It  may  be  called  then  the  effective  re- 
sistance. It  may  no  longer  be  a  constant  of  the  circuit. 

The  reactance,  x,  does  not  represent  the  expenditure  of  energy 
as  does  the  effective  resistance,  r,  but  merely  the  surging  to  and 
fro  of  energy.  It  is  not  a  constant  of  the  circuit,  but  depends 
upon  the  frequency,  and  frequently,  as  in  circuits  containing 
iron,  or  in  electrolytic  conductors,  upon  the  e.m.f.  also.  Hence 
while  the  effective  resistance,  r,  refers  to  the  power  or  active 
component  of  e.m.f.,  or  the  e.m.f.  in  phase  with  the  current,  the  re- 
actance, x,  refers  to  the  wattless  or  reactive  component  of  e.m.f., 
or  the  e.m.f.  in  quadrature  with  the  current. 

3.  The  principal  sources  of  reactance  are  electromagnetism 
and  capacity. 

Electromagnetism 

An  electric  current,  i,  in  a  circuit  produces  a  magnetic  flux 
surrounding  the  conductor  in  lines  of  magnetic  force  (or  more 
correctly,  lines  of  magnetic  induction),  of  closed,  circular,  or 
other  form,  which  alternate  with  the  alternations  of  the  current, 


INTRODUCTION  3 

and  thereby  generate  an  e.m.f.  in  the  conductor.  Since  the 
magnetic  flux  is  in  phase  with  the  current,  and  the  generated 
e.m.f.  90°,  or  a  quarter  period,  behind  the  flux,  this  e.m.f.  of 
self-induction  lags  90°,  or  a  quarter  period,  behind  the  current; 
that  is,  is  in  quadrature  therewith,  and  therefore  wattless. 

If  now  <£  =  the  magnetic  flux  produced  by,  and  interlinked 
with,  the  current,  i  (where  those  lines  of  magnetic  force  which 
are  interlinked  n-fold,  or  pass  around  n  turns  of  the  conductor, 

are  counted  n  times),  the  ratio,  — ,  is  denoted  by  L,  and  called 

% 

the  inductance  of  the  circuit.  It  is  numerically  equal,  in  absolute 
units,  to  the  interlinkages  of  the  circuit  with  the  magnetic 
flux  produced  by  unit  current,  and  is,  in  the  system  of  abso- 
lute units,  of  the  dimension  of  length.  Instead  of  the  inductance, 
L,  sometimes  its  ratio  with  the  ohmic  resistance,  r,  is  used,  and 
is  called  the  time-constant  of  the  circuit, 


If  a  conductor  surrounds  with  n  turns  a  magnetic  circuit  of 
reluctance,  (R,  the  current,  i,  in  the  conductor  represents  the 
m.m.f.  of  m  ampere-turns,  and  hence  produces  a  magnetic  flux 

777 

of  --  lines  of  magnetic  force,  surrounding  each  n  turns  of  the 
ot 

f\  ^i 

conductor,  and  thereby  giving  $  =  —  interlinkages  between 

(H 

the  magnetic   and   electric   circuits.     Hence  the  inductance  is 

L  =:  7  ==  CR' 

The  fundamental  law  of  electromagnetic  induction  is,  that 
the  e.m.f.  generated  in  a  conductor  by  a  magnetic  field  is  pro- 
portional to  the  rate  of  cutting  of  the  conductor  through  the 
magnetic  field. 

Hence,  if  i  is  the  current  and  Z/  is  the  inductance  of  a  cir- 
cuit, the  magnetic  flux  interlinked  with  a  circuit  of  current, 
it  is  Li,  and  4/Li  is  consequently  the  average  rate  of  cutting; 
that  is,  the  number  of  lines  of  force  cut  by  the  conductor  per 
second,  where  /  =  frequency,  or  number  of  complete  periods 
(double  reversals)  of  the  current  per  second,  i  —  maximum 
value  of  current. 

Since  the  maximum  rate  of  cutting  bears  to  the  average  rate 
the  same  ratio  as  the  quadrant  to  the  radius  of  a  circle  (a  sinu- 


4  ALTERNATING-CURRENT  PHENOMENA 

soidal  variation  supposed),  that  is,  the  ratio  ~  -f-  1,  the  maxi- 

2i 

mum  rate  of  cutting  is  2irf,  and,  consequently,  the  maximum 
value  of  e.m.f.  generated  in  a  circuit  of  maximum  current  value, 
i,  and  inductance,  L,  is 

e  =  2irfLi. 

Since  the  maximum  values  of  sine  waves  are  proportional  (by 
factor  \/2)  to  the  effective  values  (square  root  of  mean  squares) , 
if  i  =  effective  value  of  alternating  current,  e  =  2irfLi  is  the 

/> 

effective  value  of  e.m.f.  of  self-induction,  and  the  ratio,  -  =  2  irfL, 

is  the  inductive  reactance) 

xm  =  2  TT/L. 

Thus,  if  r  =  resistance,  xm  —  reactance,  z  =  impedance, 
the  e.m.f.  consumed  by  resistance  is  e\  —  ir\ 
the  e.m.f.  consumed  by  reactance  is  62  =  ixm; 
and,  since  both  e.m.fs.  are  in  quadrature  to  each  other,  the  total 
e.m.f.  is 

e  =  Vei2  +  e2  2  =  i  Vr2  +  xm2  =  «; 

that  is,  the  impedance,  z,  takes  in  alternating-current  circuits 
the  place  of  the  resistance,  r,  in  continuous-current  circuits. 

Capacity 

4.  If  upon  a  condenser  of  capacity  C  an  e.m.f.,  e,  is  impressed, 
the  condenser  receives  the  electrostatic  charge,  Ce. 

If  the  e.m.f.,  e}  alternates  with  the  frequency,  /,  the  average 
rate  of  charge  and  discharge  is  4  /,  and  2  irf  the  maximum  rate 
of  charge  and  discharge,  sinusoidal  waves  supposed;  hence, 
i  =  2  irfCe,  the  current  to  the  condenser,  which  is  in  quadrature 
to  the  e.m.f.  and  leading. 

It  is  then 

1 

27T/C' 

the   " condensive  reactance." 

Polarization  in  electrolytic  conductors  acts  to  a  certain  extent 
like  capacity. 

The  condensive  reactance  is  inversely  proportional  to  the 
frequency  and  represents  the  leading  out-of-phase  wave;  the 
inductive  reactance  is  directly  proportional  to  the  frequency, 
and  represents  the  lagging  out-of-phase  wave.  Hence  both  are 


INTRODUCTION  5 

of  opposite  sign  with  regard  to  each  other,  and  the  total  react- 
ance of  the  circuit  is  their  difference,  x  =  xm  —  xc. 

The  total  resistance  of  a  circuit  is  equal  to  the  sum  of  all  the 
resistances  connected  in  series;  the  total  reactance  of  a  circuit 
is  equal  to  the  algebraic  sum  of  all  the  reactances  connected 
in  series;  the  total  impedance  of  a  circuit,  however,  is  not  equal 
to  the  sum  of  all  the  individual  impedances,  but  in  general  less, 
and  is  the  resultant  of  the  total  resistance  and  the  total  reactance. 
Hence  it  is  not  permissible  directly  to  add  impedances,  as  it  is 
with  resistances  or  reactances. 

A  further  discussion  of  these  quantities  will  be  found  in  the 
later  chapters. 

5.  In  Joule's  law,  P  =  i*r,  r  is  not  the  true  ohmic  resistance, 
but  the  " effective  resistance;"  that  is,  the  ratio  of  the  power 
component  of  e.m.f.  to  the  current.     Since  in  alternating-cur- 
rent circuits,  in  addition  to  the  energy  expended  in  the  ohmic  re- 
sistance of  the  conductor,  energy  is  expended,  partly  outside, 
partly  inside  of  the  conductor,  by  magnetic  hysteresis,  mutual 
induction,   dielectric  hysteresis,   etc.,   the    effective    resistance, 
r,  is  in  general  larger  than  the  true  resistance  of  the  conductor, 
sometimes  many  time  larger,  as  in  transformers  at  open  sec- 
ondary circuit,  and  is  no  longer  a  constant  of  the  circuit.     It  is 
more  fully  discussed  in  Chapter  VIII. 

In  alternating-current  circuits  the  power  equation  contains 
a  third  term,  which,  in  sine  waves,  is  the  cosine  of  the  angle  of 
the  difference  of  phase  between  e.m.f.  and  current: 

PQ  =  ei  cos  6. 

Consequently,  even  if  e  and  i  are  both  large,  P0  may  be  very 
small,  if  cos  6  is  small,  that  is,  6  near  90°. 

KirchhofFs  laws  become  meaningless  in  their  original  form, 
since  these  laws  consider  the  e.m.fs.  and  currents  as  directional 
quantities,  counted  positive  in  the  one,  negative  in  the  opposite 
direction,  while  the  alternating  current  has  no  definite  direction 
of  its  own. 

6.  The  alternating  waves  may  have  widely  different  shapes; 
some  of  the  more  frequent  ones  are  shown  in  a  later  chapter. 

The  simplest  form,  however,  is  the  sine  wave,  shown  in  Fig.  1, 
or,  at  least,  a  wave  very  near  sine  shape,  which  may  be  repre- 
sented analytically  by 

i  =  I  sin  ^  ( t  -  tj)  =  I  sin  2irf(t  -  <i), 
to 


6 


ALTERNA  TING-C URRENT  PHENOMENA 


where  I  is  the  maximum  value  of  the  wave,  or  its  amplitude; 
tQ  is  the  time  of  one  complete  cyclic  repetition,  or  the  period  of 

the  wave,  or  /  =  —  is  the  frequency  or  number  of  complete 
to 

periods  per  second;  and  t\  is  the  time,  where  the  wave  is  zero, 
or  the  epoch  of  the  wave,  generally  called  the  phase.1 


FIG.  1. — Sine  wave. 

Obviously,  "phase"  or  "epoch"  attains  a  practical  meaning 
only  when  several  waves  of  different  phases  are  considered,  as 
"difference  of  phase."  When  dealing  with  one  wave  only,  we 
may  count  the  time  from  the  moment  when  the  wave  is  zero, 
or  from  the  moment  of  its  maximum,  representing  it  respec- 
tively by 

i  =  I  sin  2  irft, 

and  i  =  I  cos  2  irjt. 

Since  it  is  a  univalent  function  of  time,  that  is,  can  at  a  given 
instant  have  one  value  only,  by  Fourier's  theorem,  any  alter- 
nating wave,  no  matter  what  its  shape  may  be,  can  be  represented 
by  a  series  of  sine  functions  of  different  frequencies  and  different 
phases,  in  the  form 

i  =  /i  sin  2-jrf(t  -  ti)  +  72  sin  4irf(t  -  *2) 
+  /s  sin  6  wj(t  -  tt)  +    .    .    . 

where  /*,  / 2,  /s,  .  .  .  are  the  maximum  values  of  the  different 
components  of  the  wave,  ti,  £2,  ^3  ...  the  times,  where  the 
respective  components  pass  the  zero  value. 

1  "Epoch"  is  the  time  where  a  periodic  function  reaches  a  certain  value, 
for  instance,  zero;  and  "phase"  is  the  angular  position,  with  respect  to  a 
datum  position,  of  a  periodic  function  at  a  given  time.  Both  are  in  alter- 
nate-current phenomena  only  different  ways  of  expressing  the  same  thing. 


INTRODUCTION 


The  first  term,  /i  sin  2irf(t  —  £j),  is  called  the  fundamental 
wave,  or  the  first  harmonic;  the  further  terms  are  called  the  higher 
harmonics,  or  "  overtones/'  in  analogy  to  the  overtones  of  sound 
waves.  In  sin  2mrf(t  —  tn)  is  the  nih  harmonic. 

By  resolving  the  sine  functions  of  the  time  differences,  t  —  t\, 
t  —  U  .  .  .,  we  reduce  the  general  expression  of  the  wave  to 
the  form: 

i  =  A  i  sin  2  irft  -f-  A  2  sin  4  wft  +  A  3  sin  6  irft  +    .    .    . 
+  £1  cos  2  TT/Y  +  £2  cos  4  -n-ft  +  £3  cos  6  vft  +    .    .    . 

The  two  half-waves  of  each  period,  the  positive  wave  and  the 
negative  wave  (counting  in  a  definite  direction  in  the  circuit),  are 
usually  identical,  because,  for  reasons  inherent  in  their  construc- 
tion, practically  all  alternating-current  machines  generate  e.m.fs. 
in  which  the  negative  half-wave  is  identical  with  the  positive. 
Hence  the  even  higher  harmonics,  which  cause  a  difference  in 
the  shape  of  the  two  half-waves,  disappear,  and  only  the  odd 
harmonics  exist,  except  in  very  special  cases. 

Hence  the  general  alternating-current  wave  is  expressed  by: 

i  =  Ii  sin    2irf(t  -  ti)  +  73  sin  6ir/(£  -  ts) 

+  /5sin  Wirf(t  -  U)  +    .    .    ;  ' 
or, 

i  =  Ai  sin  2irft  +  A3  sin  Qirft  +  Ab  sin  10^  +    .    .    . 
+  Bi  cos  2  Trft  +  B3  cos  6  irft  -f-  B5  cos  10  irft  +    . 


\ 


FIG.  2. — Wave  without  even  harmonics. 

Such  a  wave  is  shown  in  Fig.  2,  while  Fig.  3  shows  a  wave 
whose  half -waves  are  different.  Figs.  2  and  3  represent  the  sec- 
ondary currents  of  a  Ruhmkorff  coil,  whose  secondary  coil  is 
closed  by  a  high  external  resistance;  Fig.  3  is  the  coil  operated 
in  the  usual  way,  by  make  and  break  of  the  primary  battery 


8 


ALTERNATING-CURRENT  PHENOMENA 


current;  Fig.  2  is  the  coil  fed  with  reversed  currents  by  a  com- 
mutator from  a  battery. 

7.  Inductive  reactance,  or  electromagnetic  momentum,  which 
is  always  present  in  alternating-current  circuits — to  a  large  ex- 
tent in  generators,  transformers,  etc. — tends  to  suppress  the 
higher  harmonics  of  a  complex  harmonic  wave  more  than  the 


FIG.  3. — Wave  with  even  harmonics. 

fundamental  harmonic,  since  the  inductive  reactance  is  pro- 
portional to  the  frequency,  and  is  thus  greater  with  the  higher 
harmonics,  and  thereby  causes  a  general  tendency  toward  simple 
sine  shape,  which  has  the  effect  that,  in  general,  the  alternating 
currents  in  our  light  and  power  circuits  are  sufficiently  near  sine 
waves  to  make  the  assumption  of  sine  shape  permissible. 

Hence,  in  the  calculation  of  alternating-current  phenomena, 
we  can  safely  assume  the  alternating  wave  as  a  sine  wave,  with- 
out making  any  serious  error;  and  it  will  be  sufficient  to  keep  the 
distortion  from  sine  shape  in  mind  as  a  possible  disturbing  factor, 
which,  however,  is  in  practice  generally  negligible — except  in  the 
case  of  low-resistance  circuits  containing  large  inductive  reactance 
and  large  condensive  reactance  in  series  with  each  other,  so  as  to 
produce  resonance  effects  of  these  higher  harmonics,  and  also 
under  certain  conditions  of  long-distance  power  transmission  and 
high-potential  distribution. 

S.  Experimentally,  the  impedance,  effective  resistance,  induc- 
tance, capacity,  etc.,  of  a  circuit  or  a  part  of  a  circuit  are  con- 
veniently determined  by  impressing  a  sine  wave  of  alternating 
e.m.f.  upon  the  circuit  and  measuring  with  alternating-current 


INTRODUCTION  9 

ammeter,  voltmeter  and  wattmeter  the  current,  i,  in  the  circuit, 
the  potential  difference,  e,  across  the  circuit,  and  the  power,  py 
consumed  in  the  circuit. 
Then, 

£> 

The  impedance,  z  =  -.; 

P 

The  phase  angle,  cos  6  =  —  .; 

61 
P 

The  effective  resistance,  r  =  -^. 

From  these  equations, 

The  reactance,  x  =  \/z2  —  r2. 
If  the  reactance  is  inductive,  the  inductance  is 

X 


If  the  reactance  is  condensive,  the  capacity  or  its  equivalent  is 

=  2^fx' 

wherein  /  =  the  frequency  of  the  impressed  e.m.f  .  If  the  react- 
ance is  the  resultant  of  inductive  and  condensive  reactances 
connected  in  series,  it  is 


L  and  C  can  be  found  by  measuring  the  reactance  at  two  different 
frequencies,  /i  and  /2,  as  follows; 


then, 

L  = 
and 

C  = 

A  moderate  deviation  of  the  wave  of  alternating  impressed 
e.m.f.  from  sine  shape  does  not  cause  any  serious  error  as  long  as 
the  circuit  contains  no  capacity. 

In  the  presence  of  capacity,  however,  even  a  very  slight  dis- 
tortion of  wave  shape  may  cause  an  error  of  some  hundred  per 
cent. 


10          ALTERNATING-CURRENT  PHENOMENA 

To  measure  capacity  and  condensive  reactance  by  ordinary 
alternating  currents  it  is,  therefore,  advisable  to  insert  in  series 
with  the  condensive  reactance  a  non-inductive  resistance  or  induc- 
tive reactance  which  is  larger  than  the  condensive  reactance,  or 
to  use  a  source  of  alternating  current,  in  which  the  higher  har- 
monics are  suppressed,  as  the  ^-connection  of  Constant  Potential 
— Constant-current  Transformation,  paragraph  64. 

In  iron-clad  inductive  reactances,  or  reactances  containing  iron 
in  the  magnetic  circuit,  the  reactance  varies  with  the  magnetic 
induction  in  the  iron,  and  thereby  with  the  current  and  the  im- 
pressed e.m.f.  Therefore  the  impressed  e.m.f.  or  the  magnetic 
induction  must  be  given,  to  which  the  ohmic  reactance  refers,  or 
preferably  a  curve  is  plotted  from  test  (or  calculation),  giving  the 
ohmic  reactance,  or,  as  usually  done,  the  impressed  e.m.f.  as 
function  of  the  current.  Such  a  curve  is  called  an  excitation 
curve  or  impedance  curve,  and  has  the  general  character  of  the 
magnetic  characteristic.  The  same  also  applies  to  electrolytic 
reactances,  etc. 

The  calculation  of  an  inductive  reactance  is  accomplished  by 
calculating  the  magnetic  circuit,  that  is,  determining  the  ampere- 
turns  m.m.f.  required  to  send  the  magnetic  flux  through  the 
magnetic  reluctance.  In  the  air  part  of  the  magnetic  circuit, 
unit  permeability  (or,  referred  to  ampere-turns  as  m.m.f.,  reluc- 
tivity j—  )is  used;  for  the  iron  part,  the  ampere-turns  are  taken 

4  7T/ 

from  the  curve  of  the  magnetic  characteristic,  as  discussed  in  the 
following. 


CHAPTER  II 


INSTANTANEOUS  VALUES  AND  INTEGRAL  VALUES 

9.  In  a  periodically  varying  function,  as  an  alternating  cur- 
rent, we  have  to  distinguish  between  the  instantaneous  value, 
which  varies  constantly  as  function  of  the  time,  and  the  integral 
value,  which  characterizes  the  wave  as  a  whole. 

As  such  integral  value,  almost  exclusively  the  effective  value 
is  used,  that  is,  the  square  root  of  the  mean  square ;  and  wherever 
the  intensity  of  an  electric  wave  is  mentioned  without  further 
reference,  the  effective  value  is  understood. 

The  maximum  value  of  the  wave  is  of  practical  interest  only  in 
few  cases,  and  may,  besides,  be  different  for  the  two  half-waves, 
as  in  Fig.  3. 

As  arithmetic  mean,  or  average  value,  of  a  wave  as  in  Figs.  4  and 
5,  the  arithmetical  average  of  all  the  instantaneous  values  dur- 
ing one  complete  period  is  understood. 


FIG.  4. — Alternating  wave. 

This  arithmetic  mean  is  either  =  0,  as  in  Fig.  4,  or  it  differs 
from  0,  as  in  Fig.  5.  In  the  first  case,  the  wave  is  called  an 
alternating  wave,  in  the  latter  a  pulsating  wave. 

Thus,  an  alternating  wave  is  a  wave  whose  positive  values  give 
the  same  sum  total  as  the  negative  values;  that  is,  whose  two 
half-waves  have  in  rectangular  coordinates  the  same  area,  as 
shown  in  Fig.  4. 

11 


12 


ALTERNATING-CURRENT  PHENOMENA 


A  pulsating  wave  is  a  wave  in  which  one  of  the  half-waves  pre- 
ponderates, as  in  Fig.  5. 

By  electromagnetic  induction,  pulsating  waves  are  produced 
only  by  commutating  and  unipolar  machines  (or  by  the  super- 
position of  alternating  upon  direct  currents,  etc.). 

All  inductive  apparatus  without  commutation  give  exclusively 
alternating  waves,  because,  no  matter  what  conditions  may  exist 
in  the  circuit,  any  line  of  magnetic  force  which  during  a  complete 
period  is  cut  by  the  circuit,  and  thereby  generates  an  e.m.f., 
must  during  the  same  period  be  cut  again  in  the  opposite  direc- 
tion, and  thereby  generate  the  same  total  amount  of  e.m.f.  (Ob- 
viously, this  does  not  apply  to  circuits  consisting  of  different 


AVERAGE 


VALUE 


\ 


FIG.  5. — Pulsating  wave. 

parts  movable  with  regard  to  each  other,  as  in  unipolar  machines.) 
A  direct-current  machine  without  commutator  or  collector  rings, 
or  a  coil-wound  unipolar  machine,  thus  is  an  impossibility. 

Pulsating  currents,  and  therefore  pulsating  potential  differ- 
ences across  parts  of  a  circuit  can,  however,  be  produced  from  an 
alternating  induced  e.m.f.  by  the  use  of  asymmetrical  circuits, 
as  arcs,  some  electrochemical  cells,  as  the  aluminum-carbon  cell, 
etc.  Most  of  the  alternating-current  rectifiers  are  based  on  the 
use  of  such  asymmetrical  circuits. 

In  the  following  we  shall  almost  exclusively  consider  the  alter- 
nating wave,  that  is,  the  wave  whose  true  arithmetic  mean  value 
=  0. 

Frequently,  by  mean  value  of  an  alternating  wave,  the  average 
of  one  half- wave  only  is  denoted,  or  rather  the  average  of  all 
instantaneous  values  without  regard  to  their  sign.  This  mean 
value  of  one  half-wave  is  of  importance  mainly  in  the  rectifica- 


INSTANTANEOUS  AND  INTEGRAL  VALUES       13 

tion  of  alternating  e.m.fs.,  since  it  determines  the  unidirectional 
value  derived  therefrom. 

10.  In  a  sine  wave,  the  relation  of  the  mean  to  the  maximum 
value  is  found  in  the  following  way: 

Let,  in  Fig.  6,  A  OB  represent  a  quadrant  of  a  circle  with  radius  1. 

7T 

Then,  while  the  angle  6  traverses  the  arc  ~  from  A  to  B,  the 

a 

sine  varies  from  0  to  OB  =  1.     Hence  the  average  variation  of 

7T 

the  sine  bears  to  that  of  the  corresponding  arc  the  ratio  1  -f-  ~, 

ft 

2 
or  -  -f-  1.     The  maximum  variation  of  the  sine  takes  place  about 

7T 

its  zero  value,  where  the  sine  is  equal  to  the  arc.  Hence  the 
maximum  variation  of  the  sine  is  equal  to  the  variation  of  the 


FIG.  6. 


FIG,  7. 


corresponding  arc,  and  consequently  the  maximum  variation  of 

the  sine  bears  to  its  average  variation  the  same  ratio  as  the  av- 

2 
erage  variation  of  the  arc  to  that  of  the  sine,  that  is,  IT--,  and 

since  the  variations  of  a  sine  function  are  sinusoidal  also,  we  have 

2 
Mean  value  of  sine  wave  -r-  maximum  value  =  -  -f-  1  =  0.63663. 

7T 

The  quantities,  "current,"  "e.m.f.,"  "magnetism,"  etc.,  are 
in  reality  mathematical  fictions  only,  as  the  components  of  the 
entities,  "energy,"  "power,"  etc.;  that  is,  they  have  no  inde- 
pendent existence,  but  appear  only  as  squares  or  products. 

Consequently,  the  only  integral  value  of  an  alternating  wave 
which  is  of  practical  importance,  as  directly  connected  with  the  me- 
chanical system  of  units,  is  that  value  which  represents  the  same 
power  or  effect  as  the  periodical  wave.  This  is  called  the  effective 


14 


ALTERNA  TING-C URRENT  PHENOMENA 


value.  Its  square  is  equal  to  the  mean  square  of  the  periodic 
function,  that  is: 

The  effective  value  of  an  alternating  wave,  or  the  value  repre- 
senting the  same  effect  as  the  periodically  varying  wave,  is  the  square 
root  of  the  mean  square. 

In  a  sine  wave,  its  relation  to  the  maximum  value  is  found  in 
the  following  way: 

Let,  in  Fig.  7,  AOB  represent  a  quadrant  of  a  circle  with  radius  1. 

Then,  since  the  sines  of  any  angle,  6,  and  its  complementary 
angle,  90°  —  6,  fulfill  the  condition, 

sin2  6  -f  sin2  (90  -  6}  =  1, 

the  sines  in  the  quadrant,  AOB,  can  be  grouped  into  pairs,  so 
that  the  sum  of  the  squares  of  any  pair  =  1  ;  or,  in  other  words, 
the  mean  square  of  the  sine  =  J^,  and  the  square  root  of  the 


mean  square,  or  the  effective  value  of  the  sine,  = 


That  is: 


The  effective  value  of  a  sine  function  bears  to  its  maximum  value 
the  ratio, 

J_ 

V2 
Hence,  we  have  for  the  sine  wave  the  following  relations: 


1  =  0.70711. 


Max. 

Eff. 

Arith.  mean 

Half  period 

Whole 
period 

1 

1 

V2 

2 

7T 

0 

1.0 

0.7071 

0.63663 

0 

1.4142 

1.0 

0  .  90034 

0 

1  .  5708 

1.1107 

1.0 

0 

11.  Coming  now  to  the  general  alternating  wave, 

i  =  AI  sin  2  wft  +  Az  sin  4  irft  +  As  sin  6  irft  -f-  .  .  . 
+  Bi  cos2irft  +  52  cos  4  irft  +  B3  cos  Qirft  +  .   .   ., 
we  find,  by  squaring  this  expression  and  cancelling  all  the  prod 
ucts  which  give  0  as  mean  square,  the  effective  value 


The  mean  value  does  not  give  a  simple  expression,  and  is  of  no 
general  interest. 


INSTANTANEOUS  AND  INTEGRAL  VALUES       15 

12.  All  alternating-current  instruments,  as  ammeter,  volt- 
meter, etc.,  measure  and  indicate  the  effective  value.  The  maxi- 
mum value  and  the  mean  value  can  be  derived  from  the  curve 
of  instantaneous  values,  as  determined  by  wave-meter  or 
oscillograph. 

Measurement  of  the  alternating  wave  after  rectification  by 
a  unidirectional  conductor,  as  an  arc,  gives  the  mean  value  with 
direct-current  instruments,  that  is,  instruments  employing  a 
permanent  magnetic  field,  and  the  effective  value  with  alternating- 
current  instruments. 

Voltage  determination  by  spark-gap,  that  is,  by  the  striking 
distance,  gives  a  value  approaching  the  maximum,  especially 
with  spheres  as  electrodes  of  a  diameter  larger  than  the  spark- 
gap. 


CHAPTER  III 
LAW  OF  ELECTROMAGNETIC  INDUCTION 

13.  If  an  electric  conductor  moves  relatively  to  a  magnetic 
field,  an  e.m.f.  is  generated  in  the  conductor  which  is  propor- 
tional to  the  intensity  of  the  magnetic  field,  to  the  length  of  the 
conductor,  and  to  the  speed  of  its  motion  perpendicular  to  the 
magnetic  field  and  the  direction  of  the  conductor;  or,  in  other 
words,  proportional  to  the  number  of  lines  of  magnetic  force 
cut  per  second  by  the  conductor. 

As  a  practical  unit  of  e.m.f.,  the  volt  is  defined  by  the  e.m.f. 
generated  in  a  conductor,  which  cuts  108  =  100,000,000  lines  of 
magnetic  flux  per  second. 

If  the  conductor  is  closed  upon  itself,  the  e.m.f.  produces  a 
current.  . 

A  closed  conductor  may  be  called  a  turn  or  a  convolution. 
In  such  a  turn,  the  number  of  lines  of  magnetic  force  cut  per 
second  is  the  increase  or  decrease  of  the  number  of  lines  inclosed 
by  the  turn,  or  n  times  as  large  with  n  turns. 

Hence  the  e.m.f.  in  volts  generated  in  n  turns,  or  convolutions, 
is  n  times  the  increase  or  decrease,  per  second,  of  the  flux  inclosed 
by  the  turns,  times  10~8. 

If  the  change  of  the  flux  inclosed  by  the  turn,  or  by  n  turns, 
does  not  take  place  uniformly,  the  product  of  the  number  of 
turns  times  change  of  flux  per  second  gives  the  average  e.m.f. 

If  the  magnetic  flux,  $,  alternates  relatively  to  a  number  of 
turns,  n — that  is,  when  the  turns  either  revolve  through  the 
flux  or  the  flux  passes  in  and  out  of  the  turns — the  total  flux  is 
cut  four  times  during  each  complete  period  or  cycle,  twice 
passing  into,  and  twice  out  of,  the  turns. 

Hence,  if  /  =  number  of  complete  cycles  per  second,  or  the 
frequency  of  the  flux,  $,  the  average  e.m.f.  generated  in  n  turns  is 

Eavg.  =  4  n$f  10~8  volts. 

This  is  the  fundamental  equation  of  electrical  engineering, 
and  applies  to  continuous-current,  as  well  as  to  alternating- 
current,  apparatus. 

16 


LAW  OF  ELECTROMAGNETIC  INDUCTION         17 

14.  In  continuous-current  machines  and  in  many  alternators, 
the  turns  revolve  through  a  constant  magnetic  field;  in  other 
alternators  and  in  induction  motors,  the  magnetic  field  revolves; 
in  transformers,  the  field  alternates  with  respect  to  the  sta- 
tionary turns;  in  other  apparatus,  alternation  and  rotation  occur 
simultaneously,  as  in  alternating-current  commutator  motors. 

Thus,  in  the  continuous-current  machine,  if  n  =  number  of 
turns  in  series  from  brush  to  brush,  $  =  flux  inclosed  per  turn, 
and  /  =  frequency,  the  e.m.f.  generated  in  the  machine  is 
E  =  47i<3>/10~8  volts,  independent  of  the  number  of  poles,  of 
series  or  multiple  connection  of  the  armature,  whether  of  the 
ring,  drum,  or  other  type. 

In  an  alternator  or  transformer,  if  n  is  the  number  of  turns  in 
series,  3>  the  maximum  flux  inclosed  per  turn,  and  /the  frequency, 
this  formula  gives 

Eavg.  =  4n$/10-8  volts. 

Since  the  maximum  e.m.f.  is  given  by 

7T 
•E*max.    ~  "n-^avg.j 

we  have 

Emax.  =  27m<l>/10-8volts. 

And  since  the  effective  e.m.f.  is  given  by 


e~        V2 
we  have 


=  4.44  nf&  10~8  volts, 

which  is  the  fundamental  formula  of  alternating-current  induc- 
tion by  sine  waves. 

15.  If,  in  a  circuit  of  n  turns,  the  magnetic  flux,  3>,  inclosed 
by  the  circuit  is  produced  by  the  current  in  the  circuit,  the 
ratio, 

flux  X  number  of  turns  X  10~8 
current 

is  called  the  inductance,  L,  of  the  circuit,  in  henrys. 

The  product  of  the  number  of  turns,  n,  into  the  maximum 
flux,  <£,  produced  by  a  current  of  /  amperes  effective,  or 
amperes  maximum,  is  therefore 

n$  =  L7\/2  108; 


18  ALTERNATING-CURRENT  PHENOMENA 

and  consequently  the  effective  e.m.f  .  of  self-induction  is 

E  =  \/2Trn3>flQ-* 
=  2  wfLI  volts. 

The  product,  x  =  27T/L,  is  of  the  dimension  of  resistance, 
and  is  called  the  inductive  reactance  of  the  circuit;  and  the  e.m.f. 
of  self-induction  of  the  circuit,  or  the  reactance  voltage,  is 


and  lags  90°  behind  the  current,  since  the  current  is  in  phase 
with  the  magnetic  flux  produced  by  the  current,  and  the  e.m.f. 
lags  90°  behind  the  magnetic  flux.  The  e.m.f.  lags  90°  behind 
the  magnetic  flux,  as  it  is  proportional  to  the  rate  of  change  in 
flux;  thus  it  is  zero  when  the  magnetism  does  not  change,  at  its 
maximum  value,  and  a  maximum  when  the  flux  changes  quick- 
est, which  is  where  it  passes  through  zero. 


CHAPTER  IV 
VECTOR  REPRESENTATION 

16.  While  alternating  waves  can  be,  and  frequently  are,  rep- 
resented graphically  in  rectangular  coordinates,  with  the  time  as 
abscissae,  and  the  instantaneous  values  of  the  wave  as  ordinates, 
the  best  insight  with  regard  to  the  mutual  relation  of  different 
alternating  waves  is  given  by  their  representation  as  vectors,  in 
the  so-called  crank  diagram.  A  vector,  equal  in  length  to  the 
maximum  value  of  the  alternating  wave,  revolves  at  uniform  speed 
so  as  to  make  a  complete  revolution  per  period,  and  the  pro- 
jections of  this  revolving  vector  on  the  horizontal  then  denote 
the  instantaneous  values  of  the  wave. 

Obviously,  by  this  diagram  only  sine  waves  can  be  represented 
or,  in  general,  waves  which  are  so  near  sine  shape  that  they  can  be 
represented  by  a  sine. 

Let,  for  instance,  01  represent  in  length  the  maximum  value  I  of 
a  sine  wave  of  current.  Assuming  then  a  vector,  01,  to  revolve, 
left  handed  or  in  counter-clockwise  direc- 


o 


Ai     A2 


-A 


tion,  so  that  it  makes  a  complete  revolution 
during  each  cycle  or  period  tQ.     If  then  at 
a    certain    moment    of    time,   this  vector 
stands  in  position  01  \  (Fig.  8),  the  projec- 
tion, OAi,  of  <5Z[  on  the  horizontal  line  OA 
represents  the  instantaneous  value  of  the 
current   at  this  moment.     At  a  later  mo- 
ment, O/  has  moved  farther,  to  O/2,  and  the  projection,  OAz,  of 
O/2  on  OA  is  the  instantaneous  value  at  this  later  moment.     The 
diagram  so  shows  the  instantaneous  condition  of  the  sine  wave: 
each  sine  wave  reaches  its  maximum  at  the  moment  of  time  where 
its  revolving  vector  passes  the  horizontal,  and  reaches  zero  at 
the  moment  where  its  revolving  vector  passes  the  vertical. 

If  now  the  time,  t,  and  thus  the  angle,  &  =  10 A  =  2ir  -  (where 

t0  =  time  of  one  complete  cycle  or  period)  ,Js  counted  from  the 
moment  of  time  where  the  revolving  vector  01  in  Fig.  8  stands  in 
position  O/i,  then  this  sine  wave  would  be  represented  by 

i  =  I  cos  (#  —  #1), 
19 


20  ALTERNATING-CURRENT  PHENOMENA 

where  $1  =  I\OA  may  be  called  the  phase  of  the  wave,  and 
/  =  O/i  the  amplitude  or  intensity. 

At  the  time,  #  =  $1,  that  is,  the  angle,  $1,  after  the  moment  of 
time  represented  by  position  OIi,  i  —  I,  and  OI  passes  through 
the  horizontal  OA,  that  is,  has  its  maximum  value.  The  phase 
#1  thus  is  the  angle  representing  the  time,  t\t  at  which  the  wave 
reaches  its  maximum  value. 

If  the  time,  t,  and  thus  the  angle,  #,  are  counted  from  the 
moment  at  which  the  revolving  vector  reaches  position  0/2,  the 
equation  of  the  wave  would  be 

i  =  I  cos  (#  —  #2), 

and  #2  =  IiOA  is  the  phase. 

17.  When  dealing  with  one  wave  only,  it  obviously  is  imma- 
terial from  which  moment  of  time  as  zero  value  the  time  and  thus 
the  angle,  #,  is  counted.  That  is,  the  phase  $1  or  $2  may  be  chosen 
anything  desired.  As  soon,  however,  as  several  alternating  waves 
enter  the  diagram,  it  is  obvious  that  for  all  the  waves  of  the 
same  diagram  the  time  must  be  counted  from  the  same  moment, 
and  by  choosing  the  phase  angle  of  one  of  the  waves,  that  of  the 
others  is  determined. 

Thus,  let  /  =  the  maximum  value  of  a  current,  lagging  behind 
the  maximum  value  of  voltage  E  by  time  t\, 

A    that  is,  angle  of  phase  difference  #1  =  2  TT  — 

to 

'The  phase  of  the  voltage,  E,  then  may  be 
chosen  as  a,  and  the  voltage  represented,  in 
Fig.  9,  by  vector  OE  =  E  at  phase  angle 
EOA  =  a.  As  the  current  lags  by  phase 
difference  #1,  the  phase  of  the  current  then 

must  be  fi  =  a  +  $1,  and  the  current  is  represented,  in  Fig.  9, 
by  vector  01  =  /,  under  phase  angle  /?  =  10 A. 
The  equations  of  voltage  and  current  then  are: 

e  =  E  cos  (#  —  a) 
i  =  I  cos  (#  -  /?) 

=  /  COS  (tf  -  a  -  #1). 

The  voltage  OE  =  E,  as  the  first  vector,  may  be  plotted  in  any 
desired  direction,  for  instance,  under  angle  —  a!  =  EOA  in  Fig. 
10.  The  current  then  would  be  represented  by  01  =  /,  under 


VECTOR  REPRESENTATION 


21 


phase  angle  —  (3f  =  —  (af  —  $])=  10  A,  and  the  equations  of 
voltage  and  current  would  be: 

e  =  E  cos  (#  +  «') 

i  =  I  cos  (tf  -f  £') 
=  /  cos  (tf  +  a'  -  tfi). 

Or,  the  current  01  =  I  may  be  chosen  as  the  first  vector,  in 
Fig.  9,  under  phase  angle  (3  =  10  A,  and  the 
voltage    then    would    have   the  phase  angle 
a  =  /3  —  $1,  and  be  represented  by  vector 
OE  =  E,  and  the  equations  would  be: 
i  =  I  cos  (#  —  /?) 
e  =  E  cos  (#  —  a) 
=  E  cos  (tf  -  /3  +  #1). 


FIG.  10. 


In  this  vector  representation,  a  current  lagging  behind  its 
voltage  makes  a  greater  angle  with  the  horizontal,  OA,  that  is, 
the  current  vector,  01  j  lags  behind  the  voltage  vector,  OE,  in  the 
direction  of  rotation,  thus  passes  the  zero  line,  OA,  of  maximum 
value,  at  a  later  time. 

Inversely,  a  leading  current  passes  the  zero  line  OA  earlier, 
that  is,  is  ahead  in  the  direction  of  rotation. 

Instead  of  the  maximum  value  of  the  rotating  vector,  the 
effective  value  is  commonly  used,  especially  where-  the  instan- 
taneous values  are  not  required,  but  the  diagram  intended  to 

represent  the  relations  of  the  dif- 
ferent alternating  waves  to  each 
other.  With  the  length  of  the 
rotating  vector  equal  to  the  effect- 
ive value  of  the  alternating  wave, 
the  maximum  value  obviously  is 
->-A  \/2  times  the  length  of  the  vector, 
and  the  instantaneous  values  are 
\/2  times  the  projections  of  the 
vectors  on  the  horizontal. 

18.  To  combine  different  sine 
waves,  their  graphical  representations  as  vectors,  are  combined 
by  the  parallelogram  law. 

If,  for  instance,  two  sine  waves,  OEi,  and  OEz  (Fig.  11),  are 
superposed  —  as,  for  instance,  two  e.m.fs.  acting  in  the  same  cir- 
cuit —  their  resultant  wave  is  represented  by  OE,  the  diagonal  of  a 
parallelogram  with  OEi  and  O#2  as  sides.  As  the  projection  of 


FIG.  11. 


22  ALTERNATING-CURRENT  PHENOMENA 

the  diagonal  of  a  parallelogram  equals  the  sum  of  the  projections 
of  the  sides,  during  the  rotation  of  the  parallelogram  OEiEE2, 
the  projection  of  OE  on  the  horizontal  OA,  that  is,  the  instan- 
taneous value  of  the  wave  represented  by  vector  OE,  is  equal  to 
the  sum  of  the  projection  of  the  two  sides  OEi  and  OEz,  that  is, 
the  sum  of  the  instantaneous  values  of  the  component  vectors 
0#i  and  OEz. 

From  the  foregoing  considerations  we  have  the  conclusions: 

The  sine  wave  is  represented  graphically  in  the  crank  diagram, 
by  a  vector,  which  by  its  length,  OE,  denotes  the  intensity,  and 
by  its  amplitude,  AOE,  the  phase,  of  the  sine  wave. 

Sine  waves  are  combined  or  resolved  graphically,  in  vector 
representation,  by  the  law  of  the  parallelogram  or  the  polygon  of 
sine  waves. 

KirchhofFs  laws  now  assume,  for  alternating  sine  waves,  the 
form: 

(a)  The  resultant  of  all  the  e.m.fs.  in  a  closed  circuit,  as  found 
by  the  parallelogram  of  sine  waves,  is  zero  if  the  counter  e.m.fs. 
of  resistance  and  of  reactance  are  included. 

(b)  The  resultant  of  all  the  currents  toward  a  distributing 
point,  as  found  by  the  parallelogram  of  sine  waves,  is  zero. 

The  power  equation  expressed  graphically  is  as  follows: 
The  power  of  an  alternating-current  circuit  is  represented  in 
vector  representation  by  the  product  of  the  current,  /,  into  the 
projection  of  the  e.m.f.,  E,  upon  the  current,  or  by  the  e.m.f.,  E, 

into  the  projection  of  the  current,  I, 

_  EJ_     JEO    upon  the  e.m.f.,  or  by  IE  cos  0,  where 

0  =  angle  of  phase  displacement. 

19.  Suppose,  as  an  example,  that  in 
'  a  line  having  the  resistance,  r,  and  the 
reactance,  x  =  2  irfL — where  /  =  fre- 
quency and   L  =  inductance — there 
p      12  exists  a  current  of  /  amp.,  the  line 

being   connected   to  a  non-inductive 

circuit  operating  at  a  voltage  of  E  volts.  What  will  be  the 
voltage  required  at  the  generator  end  of  the  line? 

In  the  vector  diagram,  Fig.  12,  let  the  phase  of  the  current  be 
assumed  as  the  initial  or  zero  line,  01.  Since  the  receiving  cir- 
cuit is  non-inductive,  the  current  is  in  phase  with  its  voltage. 
Hence  the  voltage,  E,  at  the  end  of  the  line,  impressed  upon  the 
receiving  circuit,  is  represented  by  a  vector,  OE.  To  overcome 


VECTOR  REPRESENTATION  23 

the  resistance,  r,  of  the  line,  a  voltage,  Ir,  is  required  in  phase 
with  the  current,  represented  by  OEi  in  the  diagram.  The 
inductive  reactance  of  the  line  generates  an  e.m.f.  which  is  pro- 
portional to  the  current,  /,  and  the  reactance,  x,  and  lags  a 
quarter  of  a  period,  or  90°,  behind  the  current.  To  overcome 
this  counter  e.m.f.  of  inductive  reactance,  a  voltage  of  the  value 
Ix  is  required,  in  phase  90°  ahead  of  the  current,  hence  represented 
by  vector  OEz*  Thus  resistance  consumes  voltage  in  phase, 
and  reactance  voltage  90°  ahead  of  the  current.  The  voltage  of 
the  generator,  EQ,  has  to  give  the  three  voltages  E,  Ei,  E2,  hence 
it  is  determined  as  their  resultant.  Combining  by  the  parallelo- 
gram law,  OEi  and  OEz,  give  OE3,  the  voltage  required  to  over- 
come the  impedance  of  the  line,  and  similarly  OEz  and  OE  give 
OEo,  the  voltage  required  at  the  generator  side  of  the  line,  to 
yield  the  voltage,  E,  at  the  receiving  end  of  the  line.  Algebraic- 
ally, we  get  from  Fig.  12 


E0  =  V(E  +  /r) 
or 


E    =  VE02  -  (Ix)2  -  Ir. 

In  this  example  we  have  considered  the  voltage  consumed  by 
the  resistance  (in  phase  with  the  current)  and  the  voltage  con- 
sumed by  the  reactance  (90°  ahead  of  the  current)  as  parts,  or 
components,  of  the  impressed  volt- 
age, EQ,  and  have  derived  EQ  by 
combining  Er,  Ex,  and  E. 

20.  We  may,  however,  introduce 
the  effect  of  the  inductive  react- 
ance directly  as  an  e.m.f.,  Er%,  the 
counter  e.m.f.  of  inductive  react- 
ance =  Ix,  and  lagging  90°  behind 
the  current;  and  the  e.m.f.  con- 
sumed  by  the  resistance  as  a 

counter  e.m.f.,  E\  =Ir,  in  opposition  to  the  current,  as  is  done 
in  Fig.  13;  and  combine  the  three  voltages  EQ,  E'i,  E'%,  to  form 
a  resultant  voltage  E,  which  is  left  at  the  end  of  the  line.  E'  \ 
and  Ef2  combine  to  form  E's,  the  counter  e.m.f.  of  impedance; 
and  since  E'z  and  EQ  must  combine  to  form  E,  EQ  is  found  as 
the  side  of  a  parallelogram,  OE0EE'3,  whose  other  side,  OE'z, 
and  diagonal  OE,  are  given. 

Or  we.  may  say  (Fig.  14),  that  to  overcome  the  counter  e.m.f. 


24 


ALTERNATING-CURRENT  PHENOMENA 


of  impedance,  OE's,  of  the  line,  the  component,  OEs,  of  the 
impressed  voltage  is  required  which,  with  the  other  component, 
OE,  must  give  the  impressed  voltage,  OE0. 

As  shown,  we  can  represent  the  voltages  produced  in  a  circuit 
in  two  ways — either  as  counter  e.m.fs.,  which  combine  with  the 
impressed  voltage,  or  as  parts,  or  components,  of  the  impressed 
voltage,  in  the  latter  case  being  of  opposite  phase.  According 
to  the  nature  of  the  problem,  either  the  one  or  the  other  way  may 
be  preferable. 


E2  — « 


Ei    0 


FIG.  14. 


As  an  example,  the  voltage  consumed  by  the  resistance  is  Ir, 
and  in  phase  with  the  current;  the  counter  e.m.f.  of  resistance  is 
in  opposition  to  the  current.  The  voltage  consumed  by  the 
reactance  is  Ix,  and  90°  ahead  of  the  current,  while  the  counter 
e.m.f.  of  reactance  is  90°  behind  the  current;  so  that,  if,  in  Fig. 
15,  OI  is  the  current. 

OEi    =  voltage  consumed  by  resistance, 
OE\  =  counter  e.m.f.  of  resistance, 
OE<z    =  voltage  consumed  by  inductive  reactance, 
=  counter  e.m.f.  of  inductive  reactance, 
=  voltage  consumed  by  impedance, 
=  counter  e.m.f.  of  impedance. 


Obviously,  these  counter  e.m.fs.  are  different  from,  for  instance, 
the  counter  e.m.f.  of  a  synchronous  motor,  in  so  far  as  they  have 
no  independent  existence,  but  exist  only  through,  and  as  long  as 
the  current  exists.  In  this  respect  they  are  analogous  to  the 
opposing  force  of  friction  in  mechanics. 

21.  Coming  back  to  the  equation  found  for  the  voltage  at  the 
generator  end  of  the  line, 


VECTOR  REPRESENTATION 


25 


we  find,  as  the  drop  of  potential  in  the  line, 
e  =  EQ  -  E  =  V(E 


Ir)2  +  (Ix)2  -  E. 
This  is  different  from,  and  less  than,  the  e.m.f.  of  impedance, 
E3  =  Iz  =  iVr2  +  x2. 

Hence  it  is  wrong  to  calculate  the  drop  of  potential  in  a  circuit 
by  multiplying  the  current  by  the  impedance;  and  the  drop  of 
potential  in  the  line  depends,  with  a  given  current  fed  over  the 
line  into  a  non-inductive  circuit,  not  only  upon  the  constants  of 
the  line,  r  and  x,  but  also  upon  the  voltage,  E,  at  the  end  of  line, 
as  can  readily  be  seen  from  the  diagrams. 

22.  If  the  receiver  circuit  is  inductive,  that  is,  if  the  current,  /, 
lags  behind  the  voltage,  E,  by  an  angle,  6,  and  we  choose  again  as 
the  zero  line,  the  current  01  (Fig.  16),  the  voltage,  OE}  is  ahead  of 


\ 


FIG.  16. 


FIG.  17. 


the  current  by  the  angle,  6.  The  voltage  consumed  by  the  resist- 
ance, Ir,  is  in  phase  with  the  current,  and  represented  by  OEi 
the  voltage  consumed  by  the  reactance,  Ix,  is  90°  ahead  of  the 
current,  and  represented  by  OE%.  Combining  OE,  OEi,  and 
OEz,  we  get  OEQ,  the  voltage  required  at  the  generator  end  of  the 
line.  Comparing  Fig.  16  with  Fig.  12,  we  see  that  in  the  former 
OE0  is  larger ;  or  conversely,  if  EQ  is  the  same,  E  will  be  less  with 
an  inductive  load.  In  other  words,  the  drop  of  potential  in  an 
inductive  line  is  greater  if  the  receiving  circuit  is  inductive  than 
if  it  is  non-inductive.  From  Fig.  16, 


E0  =  V(E  cos  0  +  Ir)2  +  (E  sin  0  +  Ix)2. 
If,  however,  the  current  in  the  receiving  circuit  is  leading,  as 


26          ALTERNATING-CURRENT  PHENOMENA 

is  the  case  when  feeding  condensers  or  synchronous  motors  whose 
counter  e.m.f.  is  larger  than  the  impressed  voltage,  then  the 
voltage  will  be  represented,  in  Fig.  17,  by  a  vector,  OE,  lagging 
behind  the  current,  Q/,  by  the  angle  of  lead,  0' ';  and  in  this  case 
we  get,  by  combining  OE  with  OEi,  in  phase  with  the  current, 
and  OEz,  90°  ahead  of  the  current,  the  generator  voltage,  OEQ, 
which  in  this  case  is  not  only  less  than  in  Fig.  16  and  in  Fig.  12, 
but  may  be  even  less  than  E',  that  is,  the  voltage  rises  in  the  line. 
In  other  words,  in  a  circuit  with  leading  current,  the  inductive 
reactance  of  the  line  raises  the  voltage,  so  that  the  drop  of  voltage 
is  less  than  with  a  non-inductive  load,  or  may  even  be  negative, 
and  the  voltage  at  the  generator  lower  than  at  the  other  end  of 
the  line. 

These  diagrams,  Figs.  12  to  17,  can  be  considered  vector  dia- 
grams of  an  alternating-current  generator  of  a  generated  e.m.f., 
EQ,  a  resistance  voltage,  E\  =  Ir,  a  reactance  voltage,  E%  =  Ix, 
and  a  difference  of  potential,  E}  at  the  alternator  terminals;  and 
we  see,  in  this  case,  that  with  the  same  generated  e.m.f.,  with  an 
inductive  load  the  potential  difference  at  the  alternator  terminals 
will  be  lower  than  with  a  non-inductive  load,  and  that  with  a 
non-inductive  load  it  will  be  lower  than  when  feeding  into  a  cir- 
cuit with  leading  current,  as  for  instance,  a  synchronous  motor 
circuit  under  the  circumstances  stated  above. 

23.  As  a  further  example,  we  may  consider  the  diagram  of  an 
alternating-current  transformer,  feeding  through  its  secondary 
circuit  an  inductive  load. 

For  simplicity,  we  may  neglect  here  the  magnetic  hysteresis, 
the  effect  of  which  will  be  fully  treated  in  a  separate  chapter  on 
this  subject. 

Let  the  time  be  counted  from  the  moment  when  the  magnetic 
flux  is  zero  and  rising.  The  magnetic  flux  then  passes  its  maxi- 
mum at  the  time  $  =  90°,  and  the  phase  of  the  magnetic  flux 
thus  is  &  =  90°,  the  flux  thus  represented  by  the  vector  0<J>  in 
Fig.  18,  vertically  downward.  The  e.m.f.  generated  by  this  mag- 
netic flux  in  the  secondary  circuit,  EI,  lags  90°  behind  the  flux; 
thus  its  vector,  OEi,  passes  the  zero  line,  OA  90°,  later  than  the 
magnetic  flux  vector,  or  at  the  time  #  =  180°;  that  is,  the  e.m.f. 
generated  in  the  secondary  by  the  magnetic  flux,  OEi,  has  the 
phase  #  =  180°.  The  secondary  current,  /i,  lags  behind  the 
e.m.f.,  EI,  by  an  angle,  0i,  which  is  determined  by  the  resistance 
and  inductive  reactance  of  the  secondary  circuit;  that  is,  by  the 


VECTOR  REPRESENTATION 


27 


load  in  the  secondary  circuit,  and  is  represented  in  the  diagram  by 
the  vector,  0Fi,  of  phase  180  +  61. 

Instead  of  the  secondary  current,  /i,  we  plot,  however,  the 
secondary  m.m.f.,  FI  =  wi/i,  where  n\  is  the  number  of  secondary 
turns,  and  FI  is  given  in  ampere-turns.  This  makes  us  inde- 
pendent of  the  ratio  of  transformation. 


FIG.  18. 

From  the  secondary  e.m.f.,  E\,  we  get  the  flux,  <i>,  required  to 
induce  this  e.m.f.,  from  the  equation 

Ei  =  \2irai/*  10  ~8; 
where 

EI  =  secondary  e.m.f.,  in  effective  volts, 
/  =  frequency,  in  cycles  per  second, 
n\  =  number  of  secondary  turns, 

$  =  maximum  value  of  magnetic  flux,  in  lines  of  magnetic 
force. 

The  derivation  of  this  equation  has  been  given  in  a  preceding 
chapter. 

This  magnetic  flux,  <3>,  is  represented  by  a  vector,  0<£,  90°  in 
phase,  and  to  produce  it  a  m.m.f.,  F,  is  required,  which  is  de- 
termined by  the  magnetic  characteristic  of  the  iron  and  the 
section  and  length  of  the  magnetic  circuit  of  the  transformer; 
this  m.m.f.  is  in  phase  with  the  flux,  <£,  and  is  represented  by  the 
vector,  OF,  in  effective  ampere-turns. 

The  effect  of  hysteresis,  neglected  at  present,  is  to  shift  OF 
ahead  of  0$,  by  an  angle,  a,  the  angle  of  hysteretic  lead.  (See 
Chapter  on  Hysteresis.) 

This  m.m.f.,  F,  is  the  resultant  of  the  secondary  m.m.f.,  F\, 


28 


ALTERNATING-CURRENT  PHENOMENA 


and  the  primary  m.m.f.,  F0;  or  graphically,  OF  is  the  diagonal  of 
a  parallelogram  with  OFi  and  OF0  as  sides.  OF\  and  OF  being 
known,  we  find  OFo,  the  primary  ampere-turns,  and  therefrom 

and  the  number  of  primary  turns,  nQ)  the  primary  current,  IQ  = 
pi 

—i  which  corresponds  to  the  secondary,  Ii. 
nQ 

To  overcome  the  resistance,  r0,  of  the  primary  coil,  a  voltage, 
j£r  =  JVo,  is  required,  in  phase  with  the  current,  Jo,  and  repre- 
sented by  the  vector,  OEr. 

To  overcome  the  reactance,  XQ  =  2irfL0,  of  the  primary  coil,  a 
voltage,  Ex  =  loXo,  is  required,  90°  ahead  of  the  current,  I0,  and 
represented  by  vector,  OEX. 

The  resultant  magnetic  flux,  <l>,  which  generates  in  the  second- 
ary coil  the  e.m.f.,  Ei,  generates  in  the  primary  coil  an  e.m.f.  pro- 

portional to 


by  the  ratio  of  turns  —  and  in  phase  with 


or, 


which  is  represented  by  the  vector,  OE'i.  To  overcome  this 
counter  e.m.f.,  E'it  a  primary  voltage,  Eit  is  required,  equal  but 
in  phase  opposition  to  E'i,  and  represented  by  the  vector,  OEi. 


The  primary  impressed  e.m.f.,  E0,  must  thus  consist  of  the 
three  components  OEi}  OEr,  and  OEXj  and  is,  therefore,  their 
resultant  OEQ,  while  the  difference  of  phase  in  the  primary  cir- 
cuit is  found  to  be 


24.  Thus,  in  Figs.  18  to  20,  the  diagram  of  a  transformer  is 
drawn  for  the  same  secondary  e.m.f.,  E\,  secondary  current,  I\t 
and  therefore  secondary  m.m.f.,  F\t  but  with  different  conditions 
of  secondary  phase  displacement: 


VECTOR  REPRESENTATION 


29 


In  Fig.  18  the  secondary  current,  /i,  lags  60°  behind  the  sec- 
ondary e.m.f.,  EI. 

In  Fig.  19,  the  secondary  current,  /i,  is  in  phase  with  the  sec- 
ondary e.m.f.,  EI. 

In  Fig.  20  the  secondary  current,  /],  leads  by  60°  the  secondary 
e.m.f.,  EI. 

These  diagrams  show  that  lag  of  the  current  in  the  secondary 
circuit  increases  and  lead  decreases  the  primary  current  and  pri- 
mary impressed  e.m.f.  required  to  produce  in  the  secondary  circuit 
the  same  e.m.f.  and  current;  or  conversely,  at  a  given  primary 
impressed  e.m.f.,  E0,  the  secondary  e.m.f.,  E\,  will  be  smaller 
with  an  inductive,  and  larger  with  a  condensive  (leading  current), 
load  than  with  a  non-inductive  load. 


FIG.  20. 

At  the  same  time  we  see  that  a  difference  of  phase  existing  in 
the  secondary  circuit  of  a  transformer  reappears  in  the  primary 
circuit,  somewhat  decreased,  if  the  current  is  leading,  and  slightly 
increased  if  lagging  in  phase.  Later  we  shall  see  that  hysteresis 
reduces  the  displacement  in  the  primary  circuit,  so  that,  with  an 
excessive  lag  in  the  secondary  circuit,  the  lag  in  the  primary 
circuit  may  be  less  than  in  the  secondary. 

A  conclusion  from  the  foregoing  is  that  the  transformer  is  not 
suitable  for  producing  currents  of  displaced  phase,  since  primary 
and  secondary  current  are,  except  at  very  light  loads,  very  nearly 
in  phase,  or  rather  in  opposition,  to  each  other. 


CHAPTER  V 
SYMBOLIC  METHOD 

25.  The  graphical  method  of  representing  alternating-current 
phenomena  affords  the  best  means  for  deriving  a  clear  insight 
into  the  mutual  relation  of  the  different  alternating  sine  waves 
entering  into  the  problem.  For  numerical  calculation,  however, 
the  graphical  method  is  generally  not  well  suited,  owing  to  the 
widely  different  magnitudes  of  the  alternating  sine  waves  rep- 
resented in  the  same  diagram,  which  make  an  exact  diagram- 
matic determination  impossible.  For  instance,  in  the  trans- 
former diagrams  (cf.  Figs.  18-20),  the  different  magnitudes  have 
numerical  values  in  practice  somewhat  like  the  following:  E\ 
=  100  volts,  and  I\  =  75  amp.  For  a  non-inductive  second- 
ary load,  as  of  incandescent  lamps,  the  only  reactance  of  the 
secondary  circuit  thus  is  that  of  the  secondary  coil,  or  x\  =  0.08 
ohms,  giving  a  lag  of  6\  =  3.6°.  We  have  also, 

n\  =      30  turns. 

n0  =    300  turns. 

FI  =  2250  ampere-turns. 

F  =100  ampere-turns. 

Er  =      10  volts. 

Ex  =      60  volts. 

Ei  =  1000  volts. 


FIG.  21. — Vector  diagram  of  transformer. 

The  corresponding  diagram  is  shown  in  Fig.  21.  Obviously, 
no  exact  numerical  values  can  be  taken  from  a  parallelogram 
as  flat  as  OFiFF0,  and  from  the  combination  of  vectors  of  the 
relative  magnitudes  1 :6  :100. 

Hence  the  importance  of  the  graphical  method  consists  not 

30 


SYMBOLIC  METHOD  31 

so  much  in  its  usefulness  for  practical  calculation  as  to  aid  in 
the  simple  understanding  of  the  phenomena  involved. 

26.  Sometimes  we  can  calculate  the  numerical  values  trigo- 
nometrically  by  means  of  the  diagram.  Usually,  however,  this 
becomes  too  complicated,  as  will  be  seen  by  trying  to  calculate, 
from  the  above  transformer  diagram,  the  ratio  of  transformation. 
The  primary  m.rn.f.  is  given  by  the  equation 


+  F1*  +  2FF1sin01, 

an  expression  not  well  suited  as  a  starting-point  for  further 
calculation. 

A  method  is  therefore  desirable  which  combines  the  exactness 
of  analytical  calculation  with  the  clearness  of  the  graphical 
representation. 

27.  We  have  seen  that  the  alternating  sine  wave  is  repre- 
sented in_  intensity,  as  well  as  phase,  by  a 
vector,  07,  which  is  determined  analytically 
by  two  numerical  quantities — the  length,  07, 
or  intensity;  and  the  amplitude,  ^4.07,  or 
phase,  6,  of  the  wave,  7.  o  a 

Instead  of  denoting  the  vector  which  repre-  FIG.  22. 

sents  the  sine  wave  in  the  polar  diagram  by 
the  polar  coordinates,  7  and  6,  we  can  represent  it  by  its  rec- 
tangular coordinates,  a  and  b  (Fig.  22),  where 

a  —  I  cos  6  is  the  horizontal  component, 

6  =  7  sin  0  is  the  vertical  component  of  the  sine  wave. 

This  representation  of  the  sine  wave  by  its  rectangular  com- 
ponents is  very  convenient,  in  so  far  as  it  avoids  the  use  of 
trigonometric  functions  in  the  combination  or  solution  of  sine 
waves. 

Since  the  rectangular  components,  a  and  b,  are  the  horizontal 
and  the  vertical  projections  of  the  vector  representing  the  sine 
wave,  and  the  projection  of  the  diagonal  of  a  parallelogram  is 
equal  to  the  sum  of  the  projections  of  its  sides,  the  combination 
of  sine  waves  by  the  parallelogram  law  is  reduced  to  the  addition, 
or  subtraction,  of  their  rectangular  components.  That  is: 

Sine  waves  are  combined,  or  resolved,  by  adding,  or  subtracting, 
their  rectangular  components. 

For  instance,  if  a  and  b  are  the  rectangular  components  of  a 
sine  wave,  7,  and  a'  and  b'  the  components  of  another  sine  wave, 


32 


ALTERNATING-CURRENT  PHENOMENA 


If  (Fig.  23),  their  resultant  sine  wave,  /o,  has  the  rectangular 
components  aQ  —  (a  +  a'),  and  bQ  =  (b  +  b'). 

To  get  from  the  rectangular  components,  a  and  6,  of  a  sine 
wave  its  intensity,  i,  and  phase,  B,  we  may  combine  a  and  b  by 
the  parallelogram,  and  derive 


Hence  we  can  analytically  operate  with  sine  waves,  as  with 

forces  in  mechanics,  by  resolving  them 
into  their  rectangular  components. 

28.  To  distinguish,  however,  the 
horizontal  and  the  vertical  com- 
ponents of  sine  waves,  so  as  not  to  be 
confused  in  lengthier  calculation,  we 
may  mark,  for  instance,  the  vertical 
components  by  a  distinguishing  index, 
°r  the  addition  of  an  otherwise  mean- 
ingless symbol,  as  the  letter  j,  and 


M 
FIG.  23. 


thus  represent  the  sine  wave  by  the  expression 


which  now  has  the  meaning  that  a  is  the  horizontal  and  b  the 
vertical  component  of  the  sine  wave  /,  and  that  both  components 
are  to  be  combined  in  the  resultant  wave  of  intensity, 


Va2  +  b2, 


and  of  phase, 


tan  0  =  — 
a 


Similarly,  a  —  jb  means  a  sine  wave  with  a  as  horizontal, 
and  —  b  as  vertical,  components,  etc. 

Obviously,  the  plus  sign  in  the  symbol,  a  +  jb,  does  not 
imply  simple  addition,  since  it  connects  heterogeneous  quan- 
tities— horizontal  and  vertical  components — but  implies  com- 
bination by  the  parallelogram  law. 

For  the  present,  j  is  nothing  but  a  distinguishing  index,  and 
otherwise  free  for  definition  except  that  it  is  not  an  ordinary 
number. 

29.  A  wave  of  equal  intensity,  and  differing  in  phase  from  the 
wave,  a  +  jb,  by  180°,  or  one-half  period,  is  represented  in 


SYMBOLIC  METHOD 


33 


polar  coordinates  by  a  vector  of  opposite  direction,  and  denoted 
by  the  symbolic  expression,  —  a  —  jb.     Or, 

Multiplying  the  symbolic  expression,  a  +  jb,  of  a  sine  wave  by 
—  1  means  reversing  the  wave,  or  rotating  it  through  180°,  or  one- 
half  period. 

A  wave  of  equal  intensity,  but  leading 
a  +  jb  by  90°,  or  one-quarter  period,  has 
(Fig.  24)  the  horizontal  component,  —  b, 
and  the  vertical  component,  a,  and  is 
represented  symbolically  by  the  expres-  FIG.  24. 

sion,  ja  —  b. 

Multiplying,  however,  a  +  jb  by  j,  we  get 


therefore,  if  we  define  the  heretofore  meaningless  symbol,  jt  by 
the  condition, 

J2  =  ~  1, 
we  have 

j(a  +  jb)  =  ja  -  b; 
hence, 

Multiplying  the  symbolic  expression,  a  +  jb,  of  a  sine  wave  by 
j  means  rotating  the  wave  through  90°,  or  one-quarter  period; 
that  is,  leading  the  wave  by  one-quarter  period. 

Similarly  — 

Multiplying   by  —  j   means    lagging   the   wave    by    one-quarter 
period. 
Since 


it  is 


and 

j  is  the  imaginary  unit,  and  the  sine  wave  is  represented  by  a 
complex  imaginary  quantity  or  general  number,  a  +  jb. 

As  the  imaginary  unit,  j,  has  no  numerical  meaning  in  the 
system  of  ordinary  numbers,  this  definition  of  j  =  \/  —  1  does 
not  contradict  its  original  introduction  as  a  distinguishing  index. 
For  the  Algebra  of  Complex  Quantities  see  Appendix  I.  For  a 
more  complete  discussion  thereof  see  "Engineering  Mathematics." 

30.  In  the  vector  diagram,  the  sine  wave  is  represented  in 
intensity  as  well  as  phase  by  one  complex  quantity, 

a  +  jb, 


34  ALTERNATING-CURRENT  PHENOMENA 

where  a  is  the  horizontal  and  6  the  vertical  component  of  the 
wave;  the  intensity  is  given  by 


*  =  V  a2  +  62, 
the  phase  by 

tan  0  =  — 
a 

and 

a  =  i  cos  6, 
b  =  i  sin  0; 

hence  the  wave,  a  -f  jb,  can  also  be  expressed  by 
i(cos  6  +  j  sin  0), 

or,  by  substituting  for  cos  6  and  sin  6  their  exponential  expres- 
sions, we  obtain 

«".> 

Since  we  have  seen  that  sine  waves  may  be  combined  or 
resolved  by  adding  or  subtracting  their  rectangular  components, 
consequently, 

Sine  waves  may  be  combined  or  resolved  by  adding  or  subtracting 
their  complex  algebraic  expressions. 
For  instance,  the  sine  waves, 

a+jb 
and 

a'  +  jb', 
combined  give  the  sine  wave, 

I  =  (a  +  o')  +j(6+-6'). 

It  will  thus  be  seen  that  the  combination  of  sine  waves  is 
reduced  to  the  elementary  algebra  of  complex  quantities. 

31.  If  /  =  i  +  jir  is  a  sine  wave  of  alternating  current,  and 
r  is  the  resistance,  the  voltage  consumed  by  the  resistance  is  in 
phase  with  the  current,  and  equal  to  the  product  of  the  current 
and  resistance.  Or 

rl  =  ri  +  jri'. 

If  L  is  the  inductance,  and  x  =  27T/L  the  inductive  react- 
ance, the  e.m.f.  produced  by  the  reactance,  or  the  counter  e.m.f. 

1  In  this  representation  of  the  sine  wave  by  the  exponential  expression  of 
the  complex  quantity,  the  angle  6  necessarily  must  be  expressed  in  radians, 
and  not  in  degrees,  that  is,  with  one  complete  revolution  or  cycle  as  2  TT,  or 

180 
with  —  =  57.3°  as  unit. 

TT 


SYMBOLIC  METHOD  35 

of  self-induction,  is  the  product  of  the  current  and  reactance, 
and  lags  in  phase  90°  behind  the  current;  it  is,  therefore,  repre- 
sented by  the  expression 

-  jxl  =  -  jxi  +  xir. 

The  voltage  required  to  overcome  the  reactance  is  consequently 
90°  ahead  of  the  current  (or,  as  usually  expressed,  the  current 
lags  90°  behind  the  e.m.f.),  and  represented  by  the  expression 

jxl  =  jxi  —  xi'. 

Hence,  the  voltage  required  to  overcome  the  resistance,  r,  and 
the  reactance,  x,  is 


that  is, 

Z  =  r  +  jx  is  the  expression  of  the  impedance  of   t  he  circuit 
in  complex  quantities. 

Hence,  if  7  =  i  -j-  ji'  is  the  current,  the  voltage  required  to 
overcome  the  impedance,  Z  =  r  +  jx,  is 

E  =  ZI  =  (r  +  jx)  (i  +  ji') 


hence,  since  j2  =  —  1 

E  =  (ri  -  xi')  +j(ri'  +  xi); 

or,  if  E  =  e  +  jef  is  the  impressed  voltage  and  Z  =  r  +  jx  the 
impedance,  the  current  through  the  circuit  is 

r  =  %  _e+je\ 
Z      r  +  jx' 

or,    multiplying   numerator   and    denominator   by   (r  —  jx)    to 
eliminate  the  imaginary  from  the  denominator,  we  have 

_  (e  -f  je')  (r  —  jx)  _  er  +  e'x       .  e'r  —  ex 
r*  +  x2  =  r2  +  x2  +  J  r*  +  x2  ' 

or,  if  E  =  e  +  jd  is  the  impressed  voltage  and  I  =  i  +  ji'  the 
current  in  the  circuit,  its  impedance  is 

E  _  e+je'       (e  +  jj)  (i  -  ji')       ei  +  e'i'        .  e'i  -  ei' 
~~ 


32.  If  C  is  the  capacity  of  a  condenser  in  series  in  a  circuit 
in  which  exists  a  current  I  =  i  -{-  ji',  the  voltage  impressed  upon 

the  terminals  of  the'  condenser  is  E  =  0    fri,  90°  behind  the  cur- 

«  £  TTJ  U 


36  ALTERNATING-CURRENT  PHENOMENA 

rent;  and  may  be  represented  by    —  0    ,n   or    —  jxj,  where 

Z  7T/U 

is   the   condensive   reactance  or   condensance   of   the 


i       o    trt 

Z  7T/O 

condenser. 

Condensive  reactance  is  of  opposite  sign  to  inductive  reactance; 
both  may  be  combined  in  the  name  reactance. 

"We  therefore  have  the  conclusion  that 
If  r  =  resistance  and  L  =  inductance, 

thus  x  =  2  TT/L  =  inductive  reactance. 

If  C  =  capacity,  Xi  =  0    fn  =  condensive  reactance, 

A  7T/O 

Z  =  r  +  j(x  —  Xi)  is  the  impedance  of  the  circuit. 
Ohm's  law  is  then  re-established  as  follows: 

E  =  ZI,    7  =  f,    Z-|- 

.  .  .  Z/  1 

The  more  general  form  gives  not  only  the  intensity  of  the  wave 
but  also  its  phase,  as  expressed  in  complex  quantities. 

33.  Since  the  combination  of  sine  waves  takes  place  by  the 
addition  of  their  symbolic  expressions,  KirchhofTs  laws  are 
now  re-established  in  their  original  form: 

(a)  The  sum  of  all  the  e.m.fs.  acting  in  a  closed  circuit  equals 
zero,  if  they  are  expressed  by  complex  quantities,  and  if  the 
resistance  and  reactance  e.m.fs.  are  also  considered  as  counter 
e.m.fs. 

(6)  The  sum  of  all  the  currents  directed  toward  a  distributing 
point  is  zero,  if  the  currents  are  expressed  as  complex  quantities. 

If  a  complex  quantity  equals  zero,  the  real  part  as  well  as  the 
imaginary  part  must  be  zero  individually;  thus,  if 

a  +  jb  =  0,  a  =  0,  b  =  0. 

Resolving  the  e.m.fs.  and  currents  in  the  expression  of  Kirch- 
hoff's  law,  we  find: 

(a)  The  sum  of  the  components,  in  any  direction,  of  all  the 
e.m.fs.  in  a  closed  circuit  equals  zero,  if  the  resistance  and 
reactance  are  represented  as  counter  e.m.fs. 

(6)  The  sum  of  the  components,  in  any  direction,  of  all  the 
currents  at  a  distributing  point  equals  zero. 

Joule's  law  and  the  power  equation  do  not  give  a  simple 
expression  in  complex  quantities,  since  the  effect  or  power  is 


SYMBOLIC  METHOD  37 

a  quantity  of  double  the  frequency  of  the  current  or  e.m.f. 
wave,  and  therefore  requires  for  its  representation  as  a  vector 
a  transition  from  single  to  double  frequency,  as  will  be  shown  in 
Chapter  XVI. 

In  what  follows,  complex  vector  quantities  will  always  be 
denoted  by  dotted  capitals  when  not  written  out  in  full;  abso- 
lute quantities  and  real  quantities  by  undotted  letters. 

34.  Referring  to  the  example  given  in  the  fourth  chapter, 
of  a  circuit  supplied  with  a  voltage,  E,  and  a  current,  /,  over  an 
inductive  line,  we  can  now  represent  the  impedance  of  the  line 
by  Z  =  r  +  jx,  where  r  =  resistance,  x  =  reactance  of  the  line, 
and  have  thus  as  the  voltage  at  the  beginning  of  the  line,  or  at 
the  generator,  the  expression 

EQ  =  E  +  ZI. 

Assuming  now  again  the  current  as  the  zero  line,  that  is, 
/  =  i,  we  have  in  general 

EQ  =  E  +  ir  +  jix; 

hence,  with  non-inductive  load,  or  E  =  e, 
EQ  =•  (e  +  ir)  +  jix, 


or  e0  =  V(e  +  ir)*  +  (IxY,       tan  00  =7r' 

In  a  circuit  with  lagging  current,  that  is,  with  leading  e.m.f., 
E  =  e  +  je',   and 

E0  =  e  +  je'  +  (r  +  jx)i 

=  (e  +  ir)  +  j(e'  +  ix), 

£/     I    ix 
or  e0  =  V(e  +  ir)*  +  (ef  +  ix)2,       tan  00  =  7+^T 

In  a  circuit  with  leading  current,  that  is,  with  lagging  e.m.f., 
E  =  e  —  je',   and 

E0=  (e-  je')  +  (r  +  jx)i 

=  (e  +  ir)  -  j(e'  -  ix), 

e'  —  ix 
or          60  =  V(e+'ir)*+  (e'  -  ix)\       tan  eQ  =  -   e  +  ir> 

values  which  easily  permit  calculation. 

35.  When  transferring  from  complex  quantities  to  absolute 
values,  it  must  be  kept  in  mind  that: 

The  absolute  value  of  a  product  or  a  ratio  of  complex  quanti- 
ties is  the  product  or  ratio  of  their  absolute  values. 


38          ALTERNATING-CURRENT  PHENOMENA 

The  phase  angle  of  a  product  or  a  ratio  of  complex  quantities 
is  the  sum  or  difference  of  their  phase  angles. 

That  is,  if 

A  '  =  a'  +  jV  =  a(cos  a  +  j  sin  a) 
B  =  V  +  jb"  =  6(cos  0  +  jf  sin  0) 
C  =  c'  +  jc"  =  c(cos  7  +  j  sin  7) 


.  ab 

the  absolute  value  of  -77-  is  given  by  —  >  and  its  phase  angle  by 
o  c 

a  -j-  0  —  7,  that  is,  it  is 

AB       ab 

-g-  =  ^[cos  (a  +  0  -  7)  +  j  sin  (a  +  0  -  7)], 

where 


a  =  Va/2  +  a"2 


c  =  Vc/2  +  c//2 

are  the  absolute  values  of  A,  B  and  C. 

This  rule  frequently  simplifies  greatly  the  derivation  of  the 
absolute  value  and  phase  angle,  from  a  complicated  complex 
expression. 


CHAPTER  VI 
TOPOGRAPHIC  METHOD 

36.  In  the  representation  of  alternating  sine  waves  by  vectors, 
a  certain  ambiguity  exists,  in  so  far  as  one  and  the  same  quantity 
—voltage,  for  instance — can  be  represented  by  two  vectors  of 
opposite  direction,  according  as  to  whether  the  e.m.f.  is  considered 
as  a  part  of  the  impressed  voltage  or  as  a  counter  e.m.f.  This  is 
analogous  to  the  distinction  between  action  and  reaction  in 
mechanics. 

Further,  it  is  obvious  that  if  in  the  circuit  of  a  generator,  G 
(Fig.  25),  the  current  in  the  direction  from  terminal  A  over  re- 
sistance R  to  terminal  B  is  represented  by  a  vector,  01  (Fig.  26), 
or  by  7  =  i  +  ji',  the  same  current  can  be  considered  as  being 


FIG.  25. 


FIG.  26. 


in  the  opposite  direction,  from  terminal  B  to  terminal  A  in  op- 
posite phase,  and  therefore  represented  by  a  vector,  01 1  (Fig.  26), 
or  by  1 1  =  —  i  —  ji' . 

Or,  if  the  difference  of  potential  from  terminal  B  to  terminal 
A  is  denoted  by  the  E  =  e  +  jef,  the  difference  of  potential  from 
A  to  B  is  Ei  =  —  e  —  je'. 

Hence,  in  dealing  with  alternating-current  sine  waves  it  is 
necessary  to  consider  them  in  their  proper  direction  with  regard 
to  the  circuit.  Especially  in  more  complicated  circuits,  as  inter- 
linked polyphase  systems,  careful  attention  has  to  be  paid  to 
this  point. 

37.  Let,  for  instance,  in  Fig,  27,  an  interlinked  three-phase 
system  be  represented  diagrammatically  as  consisting  of  three 

39 


40  ALTERNA  TING-C  URRENT  PHENOMENA 

voltages,  of  equal  intensity,  differing  in  phase  by  one-third  of  a 
period.  Let  the  voltages  in  the  direction  from  the  common  con- 
nection, 0,  of  the  three  branch  circuits  to  the  terminals,  AI,  A2, 
A3,  be  represented  by  EI,  E2,  E3.  Then  the  difference  of  poten- 
tial from  A 2  to  A  i  is  E2  —  EI,  since  the  two  voltages,  EI  and  E2, 
are  connected  in  circuit  between  the  terminals,  AI  and  A2,  in  the 
direction  AI — 0 — A2;  that  is,  the  one,  E2,  in  the  direction,  OA2, 
from  the  common  connection  to  terminal,  the  other,  EI,  in  the 
opposite  direction,  AiO,  from  the  terminal  to  common  connec- 
tion, and  represented  by  —  EI.  Conversely,  the  difference  of 
potential  from  AI  to  A2  is  EI  —  E2. 

It  is  then  convenient  to  go  still  a  step  farther,  and  drop  the 
vector  line  altogether  in  the  diagrammatic  representation;  that 
is,  denote  the  sine  wave  by  a  point  only,  the  end  of  the  corre- 
sponding vector. 

Looking  at  this  from  a  different  point  of  view,  it  means  that 
we  choose  one  point  of  the  system — for  instance,  the  common 


OEl 


O 

Ex 

FIG.  27.  FIG.  28. 

connection,  or  neutral  0 — as  a  zero  point,  or  point  of  zero  poten- 
tial, and  represent  the  potentials  of  all  the  other  points  of  the 
circuit  by  points  in  the  diagram,  such  that  their  distances  from 
the  zero  point  give  the  intensity,  their  amplitude  the  phase  of 
the  difference  of  potential  of  the  respective  point  with  regard  to 
the  zero  point;  and  their  distance  and  amplitude  with  regard  to 
other  points  of  the  diagram,  their  difference  of  potential  from 
these  points  in  intensity  and  phase. 

Thus,  for  example,  in  an  interlinked  three-phase  system  with 
three  voltages  of  equal  intensity,  and  differing  in  phase  by  one- 
third  of  a  period,  we  may  choose  the  common  connection  of  the 
star-connected  generator  as  the  zero  point,  and  represent,  in 
Fig.  28,  one  of  the  voltages,  or  the  potential  at  one  of  the  three- 


TOPOGRAPHIC  METHOD 


41 


phase  terminals,  by  point  E\.  The  potentials  at  the  two  other 
terminals  will  then  kbe  given  by  the  points  E%  and  Es,  which  have 
the  same  distance  from  0  as  EI,  and  are  equidistant  from  E\  and 
from  each  other. 

The  difference  of  potential  between  any  pair  of  terminals,  for 
instance,  EI  and  E%,  is  then  thp  distance  ^2^1,  or  EiE%,  according 
to  the  direction  considered. 

38.  If  now  the  three  branches,  OEi,  OE2  and  OEl,  of  the 
three-phase  system  are  loaded  equally  by  three  currents  equal 
in  intensity  and  in  difference  of  phase  against  their  voltages, 


Eo 
a 


BALANCED  THREE-PHASE  SYSTEM1 
NON-INDUCTIVE  LOAD 


FIG.  29. 


FIG.  30. 


these  currents  are  represented  in  Fig.  29  by  the  vectors  0/i  = 
01 2  =  01 3  =  J,  lagging  behind  the  voltages  by  angles  E\0l\  = 
#20/2  =  #30/3  =  8. 

Let  the  three-phase  circuit  be  supplied  over  a  line  of  impedance, 
Zi  =  7*1  +  jxi,  from  a  generator  of  internal  impedance,  ZQ  = 

XQ  +  JXQ. 

In  phase  OE\  the  voltage  consumed  by  resistance  r\  is  repre- 
sented by  thejdistance,  EiEJ  =  Iri,  in  phase,  that  is,  parallel 
with  current  01 1.  The  voltage  consumed  by  reactance  x\  is 
represented  by  E^Ei11  =  Ixi,  90°  ahead  of  current  0/i.  The 
same  applies  to  the  other  two  phases,  and  it  thus  follows  that  to 
produce  the  voltage  triangle,  EiE2Es,  at  the  terminals  of  the 
consumer's  circuit,  the  voltage  triangle,  EinE2llE3n,  is  required 
at  the  generator  terminals. 


42 


ALTERNATING-CURRENT  PHENOMENA 


Repeating  the  same  operation  for  the  internal  impedance  of 
the  generator,  we  get  EllEni  =  Jr0,  and  parallel  to  0/i,  EU1E°  = 
Ix0,  and  90°  ahead  of  0/i,  and  thus  as  triangle  of  (nominal)  gen- 
erated e.m.fs.  of  the  generator,  Ei°EzQEs°. 

In  Fig.  29  the  diagram  is  shown  for  45°  lag,  in  Fig.  30  for  non- 
inductive  load,  and  in  Fig.  31  for  45°  lead  of  the  currents  with 
regard  to  their  voltages. 

As  seen,  the  generated  e.m.f.  and  thus  the  generator  excitation 
with  lagging  current  must  be  higher,  and  with  leading  current 
lower,  than  at  non-inductive  load,  or  conversely  with  the  same 
generator  excitation,  that  is,  the  same  internal  generator  e.m.f. 


SINGLE-PHASE  CIRCUIT 

60°  LAG 

CABLE  OF  DISTRIBUTED 
CAPACITY  AND  RESISTANCE 


FIG.  32. 


triangle,  EiQE^EzQ,  the  voltages  at  the  receiver's  circuit,  E\,  Ez, 
Es,  fall  off  more  with  lagging,  and  less  with  leading  current,  than 
with  non-inductive  load. 

39.  As  a  further  example  may  be  considered  the  case  of  a 
single-phase  alternating-current  circuit  supplied  over  a  cable 
containing  resistance  and  distributed  capacity. 

Let,  in  Fig.  32,  the  potential  midway  between  the  two  ter- 
minals be  assumed  as  zero  point  0.  The  two  terminal  voltages 
at  the  receiver  circuit  are  then  represented  by  the  points  E  and 
E1,  equidistant  from  0  and  opposite  each  other,  and  the  two  cur- 
rents at  the  terminals  are  represented  by  the  points  /  and  71, 
equidistant  from  0  and  opposite  each  other,  and  under  angle  0 
with  E  and  E1  respectively. 

Considering  first  an  element  of  the  line  or  cable  next  to  the 
receiver  circuit.  In  this  voltage,  EEi,  is  consumed  byjthe  re- 
sistance of  the  line  element,  in  phase  with  the  current,  O/,  and 
proportional  thereto,  and  a  current,  77~i,  consumed  by  the 


TOPOGRAPHIC  METHOD  43 

capacity,  as  charging  current  of  the  line  element,  90°  ahead  in 
phase  of  the  voltage,  OE,  and  proportional  thereto,  so  that  at  the 
generator  end  of  this  cable  element  current  and  voltage  are  01 \ 
and  OEi  respectively. 

Passing  now  to  the  next  cable  element  we  have  again  a  voltage, 
EiEzj  proportional  to  and  in  phase  with  the  current,  O/i,  and  a 
current,  /i/2,  proportional  to  and  90°  ahead  of  the  voltage,  OEi, 
and  thus  passing  from  element  to  element  along  the  cable  to  the 
generator,  we  get  curves  of  voltages,  e  and  e1,  and  curves  of  cur- 
rents, i  and  il,  which  can  be  called  the  topographical  circuit 
characteristics,  and  which  correspond  to  each  other,  point  for 
point,  until  the  generator  terminal  voltages,  OEo  and  OEo1,  and 
the  generator  currents,  O/o  and  O/o1,  are  reached. 

Again,  adding  EQEn  =  70r0  and  parallel  to  OTi  and  EnE°  = 
IQx0  and  90°  ahead  of  0/0,  gives  the  (nominal)  generated  e.m.f. 
of  the  generator  OE°,  where  ZQ  =  r0  +  jxQ  =  internal  impedance 
of  the  generator. 

In  Fig.  32  is  shown  the  circuit  characteristics  for  60°  lag  of 
a  cable  containing  only  resistance  and  capacity. 

Obviously  by  graphical  construction  the  circuit  characteristics 
appear  more  or  less  as  broken  lines,  due  to  the  necessity  of  using 
finite  line  elements,  while  in  reality  they  are  smooth  curves  when 
calculated  by  the  differential  method,  as  explained  in  Section 
III  of  "  Theory  and  Calculation  of  Transient  Electric  Phenomena 
and  Oscillations." 

40.  As  further  example  may  be  considered  a  three-phase  cir- 
cuit supplied  over  a  long-distance  transmission  line  of  distrib- 
uted capacity,  self-induction,  resistance,  and  leakage. 

Let,  in  Fig.  33,  OE1}  OEz,  OEs  =  three-phase  voltages  at  re- 
ceiver circuit,  equidistant  from  each  other  and  =  E. 

Let  O/i,  0/2,  0/3  =  three-phase  currents  in  the  receiver  cir- 
cuit equidistant  from  each  other  and  =  /,  and  making  with  E 
the  phase  angle,  0. 

Considering  again  as  in  §3  the  transmission  line,  element  by 
element,  we  have  in  every  element  a  voltage,  EiEi1,  consumed 
by  the  resistance  in  phase  with  the  current,  O/i,  and  proportional 
thereto,  and  a  voltage,  Ei1,  Eili,  consumed  by  the  reactance  of 
the  line  element,  90°  ahead  of  the  current,  O/i,  and  proportional 
thereto.  

In  the  same  line  element  we  have  a  current,  /i/i1,  in  phase 
with  the  voltage,  OEi,  and  proportional  thereto,  representing 


44 


ALTERNATING-CURRENT  PHENOMENA 


the  loss  of  current  by  leakage,  dielectric  hysteresis,  etc.,  and  a 
current,  /i1  /i11,  90°  ahead  of  the  voltage,  OEi,  and  proportional 
thereto,  the  charging  current  of  the  line  element  as  condenser; 
and  in  this  manner  passing  along  the  line,  element  by  element, 
we  ultimately  reach  the  generator  terminal  voltages,  EI°,  E2°,  EZ°, 


10 


THREE  PHASE  CIRCUIT 

60  LAG 

TRANSMISSION  LINE 

WITH  DISTRIBUTED 

CAPACITY,  INDUCTANCE 

RESISTANCE.  AND  LEAKAGE 


FIG.  33. 


TRANSMISSION 

WITH  DISTRIBUTED 

CAPACITY,  INDUCTANCE 

RESISTANCE  AND  LEAKAGE 

90°  LAG 


FIG.  34. 


and  generator  currents,  /i°,  72°,  /s°,  over  the  topographical  char- 
acteristics of  voltage,  ei,  £2,  e3,  and  of  current,  i\,  iz,  is,  as  shown 
in  Fig.  33. 

The  circuit  characteristics  of  current,  i,  and  of  voltage,  e,  cor- 
respond to  each  other,  point  for  point,  the  one  giving  the  current 
and  the  other  the  voltage  in  the  line  element. 

Only  the  circuit  characteristics  of  the  first  phase  are  shown, 


TOPOGRAPHIC  METHOD 


45 


as  ei  and  i\.  As  seen,  passing  from  the  receiving  end  toward 
the  generator  end  of  the  line,  potential  and  current  alternately 
rise  and  fall,  while  their  phase  angle  changes  periodically  be- 
tween lag  and  lead. 

41.  More  markedly  this  is  shown  in  Fig.  34,  the  topographic 
circuit  characteristic  of  one  of  the  lines  with  90°  lag  in  the  receiver 
circuit.  Corresponding  points  of  the  two  characteristics,  e  and 
i,  are  marked  by  corresponding  figures  0  to  16,  representing  equi- 
distant points  of  the  line.  The  values  of  voltage,  current  and 


TRANSMISSION  LINE 

WITH  DISTRIBUTED  CAPACITY,  INDUCTANCE 
RESISTANCE  AND  LEAKAGE 


FIG.  35. 

their  difference  of  phase  are  plotted  in  Fig.  35  in  rectangular 
coordinates  with  the  distance  as  abscissas,  counting  from  the 
receiving  circuit  toward  the  generator.  As  seen  from  Fig.  35, 
voltage  and  current  periodically  but  alternately  rise  and  fall, 
a  maximum  of  one  approximately  coinciding  with  a  minimum 
of  the  other,  and  with  a  point  of  zero  phase  displacement.  The 
phase  angle  between  current  and  e.m.f.  changes  from  90°  lag 
to  72°  lead,  44°  lag,  34°  lead,  etc.,  gradually  decreasing  in  the 
amplitude  of  its  variation. 


CHAPTER  VII 
POLAR  COORDINATES  AND  POLAR  DIAGRAMS 

42.  The  graphic  representation  of  alternating  waves  in  rec- 
tangular coordinates,  with  the  time  as  abscissae  and  the  instan- 
taneous values  as  ordinates,  gives  a  picture  of  their  wave  structure, 
as  shown  in  Figs.   1  to  5.     It  does  not,  however,  show  their 
periodic  character  as  well  as  the  representation  in  polar  coordi- 
nates, with  the  time  as  the  angle  or  the  amplitude — one  complete 
period  being  represented  by  one  revolution — and  the  instan- 
taneous values  as  radius  vectors;  the  polar  coordinate  system, 
in  which  the  independent  variable,  the  angle,  is  periodic,  obvi- 
ously lends  itself  better  to  the  representation  of  periodic  functions, 
as  alternating  waves. 

Thus  the  two  waves  of  Figs.  2  and  3  are  represented  in  polar 
coordinates  in  Figs.  36  and  37  as  closed  characteristic  curves, 
which,    by   their  intersection  with  the  radius 
vector,    give    the   instantaneous  value   of  the 
wave,   corresponding  to  the  time  represented 
by  the  amplitude  or  angle  of  the  radius  vector. 
These  instantaneous  values  are  positive  if  in 
the  direction  of  the  radius  vector,  and  negative 
if  in  opposition.       Hence  the  two  half-waves 
.   in  Fig.  2  are  represented  by  the  same  polar 
characteristic  curve,  which  is  traversed  by  the 
point  of  intersection  of  the  radius  vector  twice 

T71      _          OC 

per  period — once  in  the  direction  of  the  vector, 
giving  the  positive  half-wave,  and  once  in  opposition  to  the 
vector,  giving  the  negative  half-wave.  In  Figs.  3  and  37 
where  the  two  half-waves  are  different,  they  give  different  polar 
characteristics. 

43.  The  sine  wave,  Fig.  1,  is  represented  in  polar  coordinates 
by  one  circle,  as  shown  in  Fig.  38.     The  diameter  of  the  char- 
acteristic curve  of  the  sine  wave,  /  =  OC,  represents  the  intensity 
of  the  wave;  and  the  amplitude  of  the  diameter  OC,  ^C  0o  =  AOC, 
is  the  phase  of  the  wave,  which,  therefore,  is  represented  analytic- 
ally by  the  function 

i  =  I  cos  (B  -  60), 
46 


POLAR  COORDINATES  AND  POLAR  DIAGRAMS    47 

where  6  =  2  TT  —  is   the   instantaneous   value   of   the   amplitude 

*o 

corresponding  to  the  instantaneous  value,  i,  of  the  wave. 

The  instantaneous  values  are  cut  out  on  the  movable  radius 
vector  by  its  intersection  with  the  characteristic  circle.     Thus, 

for  instance,  at  the  amplitude,  AOBi  =  61  =  ^TT—  (Fig.  38),  the 

to 

instantaneous    value  is  OB'-}  at  the  amplitude,  AOB2  =  62  = 

*  , 

2  ir^j  the  instantaneous  value  is  OB",  and   negative,  since  in 
to 

opposition  to  the  radius  vector,  OB%. 

The  angle,  0,  so  represents  the  time,  and  increasing  time  is 
represented  by  an  increase  of  angle  6  in  counter-clockwise  rota- 


FIG.  37. 


tion.  That  is,  the  positive  direction,  or  increase  of  time,  is 
chosen  as  counter-clockwise  rotation,  in  conformity  with  general 
custom. 

The  characteristic  circle  of  the  alternating  sine  wave  is  deter- 
mined by  the  length  of  its  diameter — the  intensity  of  the  wave; 
and  by  the  amplitude  of  the  diameter — the  phase  of  the  wave. 

Hence  wherever  the  integral  value  of  the  wave  is  considered 
alone,  and  not  the  instantaneous  values,  the  characteristic  circle 
may  be  omitted  altogether,  and  the  wave  represented  in  intensity 
and  in  phase  by  the  diameter  of  the  characteristic  circle. 

Thus,  in  polar  coordinates,  the  alternating  wave  may  be  repre- 
sented in  intensity  and  phase  by  the  length  and  direction  of  a 
vector,_OC,  Fig.  38,  and  its  analytical  expression  would  then  be 
c  =  OC  cos  (0  -  00). 

This  leads  to  a  second  vector  representation  of  alternating 


ALTERNATING-CURRENT  PHENOMENA 


waves,  differing  from  the  crank  diagram  discussed  in  Chapter  IV. 
It  may  be  called  the  time  diagram  or  polar  diagram,  and  is  used 
to  a  considerable  extent  in  the  literature,  thus  must  be  familiar 
to  the  engineer,  though  in  the  following  we  shall  in  graphic 
representation  and  in  the  symbolic  representation  based  thereon, 
use  the  crank  diagram  of  Chapters  IV  and  V. 

In  the  time  diagram  as  well  as  in  the  crank  diagram,  instead 
of  the  maximum  value  of  the  wave,  the  effective  value,  or  square 
root  of  mean  square,  may  be  used  as  the  vector,  which  is  more 
convenient ;  and  the  maximum  value  is  then  \/2  times  the  vector 
OC,  so  that  the  instantaneous  values,  when  taken  from  the  dia- 
gram, have  to  be  increased  by  the  factor  \/2. 

Thus,  the  wave, 

b  =  B  cos  2irf(t  -  ti) 
=  B  cos  (0  -  0i), 

A    is,  in  Fig.  39,  represented  by 

T> 

vector  OB  = 


c        of  phase 

and  the  wave, 


FIG.  39. 
is,  in  Fig.  39,  represented  by 


AOB  =  0!j 

c  =  C  cos2irf(t  +  tz) 
=  C  cos  (0  -f  02) 


£ 

vector  OC  = 


of  phase 


V2 
AOC  =  -  02. 


The  former  is  said  to  lag  by  angle  0i,  the  latter  to  lead  by  angle 
02,  with  regard  to  the  zero  position. 

The  wave  b  lags  by  angle  (0i  +  02)  behind  wave  c,  or  c  leads 
b  by  angle  (0i  -f  02). 

44.  To  combine  different  sine  waves,  their  graphical  repre- 
sentations, or  vectors,  are  combined  by  the  parallelogram  law. 

From  the  foregoing  considerations  we  have  the  conclusions: 

The  sine  wave  is  represented  graphically  in  polar  coordinates 
by  a  vector,  which  by  its  length  OC,  denotes  the  intensity,  and  by 
its  amplitude,  AOC,  the  phase,  of  the  sine  wave. 

Sine  waves  are  combined  or  resolved  graphically,  in  polar 
coordinates,  by  the  law  of  the  parallelogram  or  the  polygon  of 
sine  waves.  (Fig.  40.) 


POLAR  COORDINATES  AND  POLAR  DIAGRAMS    49 


Kirchhoffs  laws  now  assume,  for  alternating  sine  waves,  the 
form: 

(a)  The  resultant  of  all  the  e.m.fs.  in  a  closed  circuit,  as  found 
by  the  parallelogram  of  sine  waves,  is  zero  if  the  counter  e.m.fs. 
of  resistance  and  of  reactance  are  included. 

(b)  The    resultant    of    all    the 
currents    toward    a    distributing 
point,  as  found  by  the  parallelo- 
gram of  sine  waves,  is  zero. 

The  power  equation  expressed 
graphically  is  as  follows: 

The  power  of  an  alternating- 
current  circuit  is  represented  in 
polar  coordinates  by  the  product 
of  the  current,  /,  into  the  projec- 
tion of  the  e.m.f.,  E,  upon  the 
current,  or  by  the  e.m.f.,  E,  into  the  projection  of  the  current, 
/,  upon  the  e.m.f.,  or  by  IE  cos  6,  where  9  =  angle  of  time- 
phase  displacement. 

45.  The  instances  represented  by  the  vector  representation  of 
the  crank  diagram  in  Chapter  IV  as  Figs.   16,   17,  18,  19,  20, 


FIG.  40. 


FIG.  41. 


FIG.  42. 


then  appear  in  the  vector  representation  of  the  time  diagram  or 
polar  coordinate  diagram,  in  the  form  of  Figs.  41,  42,  43, 
44,  45. 

These  figures  are  the  reverse,  or  mirror  image  of  each  other. 
That  is,  the  crank  diagrams,  turned  around  the  horizontal  (or 
any  other  axis) ,  so  as  they  would  be  seen  in  a  mirror,  are  the  time 
diagrams,  and  inversely. 


50        -  ALTERNATING-CURRENT  PHENOMENA 

The  polar  diagram,  Fig.  46,  of  a  current: 

i  =  I  cos  (&  -  &) 
represented  by  vector  07, 


FIG.  43. 


FIG.  45. 

lagging  behind  the  voltage : 

e  =  E  cos  (#  —  a) 
represented  by  vector  OE, 
by  angle 

0i  =  ft  -  a 
then  means: 


FIG.  46. 


POLAR  COORDINATES  AND  POLAR  DIAGRAMS   51 

The  voltage  e  reaches  its  maximum  at  the  time  t\t  which  is 

represented  by  angle  a  =  2  ir-~>  where  tQ  =  period,  and  the  cur- 

to 

rent,  i,  reaches  its  maximum  at  the  time  tZ)  which  is  represented  by 

angle  0  =  2  TT~,  and  since  /5  >  a,  the  current  reaches  its  maximum 
to 

at  a  later  time  than  the  voltage,  that  is,  lags  behind  the  voltage, 
and  the  lag  of  the  current  behind  the  voltage  is  the  difference 
between  the  times  of  their  maxima,  /3  and  a,  in  angular  measure, 
that  is,  is 


At  any  moment  of  time  t,  represented  by  angle  6  =  2  TT—  >  the  in- 

fo 

stantaneous  values  of  current  and  voltage,  i  and  e,  are  the  projec- 
tions of  01  and  OE  on  the  time  radius  OX  drawn  under  angle 
AOX  =  B. 

The  crank  diagram  corresponding  to  the  time  diagram  Fig. 
46  is  shown  in  Fig.  47.  It  means:  The  vectors  01  and  OE, 
representing  the  current  and  the  voltage  respectively,  rotate 
synchronously,  and  by  their  projections  on  the  horizontal  OA 
represent  the  instantaneous  values  of  current  and  voltage. 
Angle  10  A  =  /3  being  larger  than  angle  EOA  =  a,  the  current 
vector  01  passes  its  maximum,  in  position  OA,  later  than  the 
voltage  vector  OE,  that  is,  the  current  lags  behind  the  voltage, 
by  the  difference  of  time  corresponding  to  the  passage  of  the 
current  and  voltage  vectors  through  their  maxima,  in  the  direc- 
tion OA,  that  is,  by  the  time  angle  0i  =  /3  —  a. 

A  polar  diagram,  Fig.  46,  with  the  current,  01,  lagging  behind 
the  voltage,  OE,  by  the  angle,  0i,  thus  considered  as  crank  dia- 
gram would  represent  the  current  leading  the  voltage  by  the 
angle,  0i,  and  a  crank  diagram,  Fig.  47,  with  the  current  lagging 
behind  the  voltage  by  the  angle,  0i,  would  as  polar  diagram 
represent  a  current  leading  the  voltage  by  the  angle,  0i. 

46.  The  main  difference  in  appearance  between  the  crank  dia- 
gram and  the  polar  diagram  therefore  is  that,  with  the  same 
direction  of  rotation,  lag  in  the  one  diagram  is  represented  in  the 
same  manner  as  lead  in  the  other  diagram,  and  inversely.  Or, 
a  representation  by  the  crank  diagram  looks  like  a  representation 
by  the  polar  diagram,  with  reversed  direction  of  rotation,  and 
vice  versa.  Or,  the  one  diagram  is  the  image  of  the  other  and  can 


52  ALTERNATING-CURRENT  PHENOMENA 

be  transformed  into  it  by  reversing  right  and  left,  or  top  and 
bottom.  So  the  crank  diagram,  Fig.  47,  is  the  image  of  the  polar 
diagram,  Fig.  46. 

In  symbolic  representation,  based  upon  the  crank  diagram,  the 
impedance  was  denoted  by 

Z  =  r  +  jx, 
where  x  =  inductive  reactance. 

In  the  polar  diagram,  the  impedance  thus  is  denoted  by: 

Z  =  r  -  jx 

since  the  latter  is  the  mirror  image  of  the  crank  diagram,  that  is, 
differs  from  it  symbolically  by  the  interchange  of  +  j  and  —  j. 

A  treatise  written  in  the  symbolic  repre- 
sentation by  the  polar  diagram,  thus  can  be 
translated  to  the  representation  by  the  crank 
diagram,  and  inversely,  by  simply  reversing 
the  signs  of  all  imaginary  quantities,  that  is, 
considering  the  signs  of  all  terms  with  j 

FIG.  47.  changed 

A  graphical  representation  in  the  polar  dia- 
gram can  be  considered  as  a  graphic  representation  in  the  crank 
diagram,  with  clockwise  or  right-handed  rotation,  and  inversely. 

Thus,  for  the  engineer  familiar  with  one  representation  only,  but 
less  familiar  with  the  other,  the  most  convenient  way  when  meet- 
ing with  a  treatise  in  the,  to  him,  unfamiliar  representation  is  to 
consider  all  the  diagrams  as  clockwise  and  all  the  signs  of  j  reversed. 

In  conformity  with  the  recommendation  of  the  Turin  Congress 
— however  ill  considered  this  may  appear  to  many  engineers — in 
the  following  the  crank  diagram  will  be  used,  and  wherever 
conditions  require  the  time  diagram,  the  latter  be  translated  to 
the  crank  diagram.  It  is  not  possible  to  entirely  avoid  the  time 
diagram,  since  the  crank  diagram  is  more  limited  in  its  application. 

47.  The  crank  diagram  offers  the  disadvantage,  that  it  can  be 
applied  to  sine  waves  only,  while  the  polar  diagram  permits  the 
construction  of  the  curve  of  waves  of  any  shapes,  as  those  in 
Figs.  36  and  37. 

In  most  cases,  this  objection  is  not  serious,  and  in  the  diagram- 
matic and  symbolic  representation,  the  alternating  quantities 
can  be  assumed  as  sine  waves,  that  is,  the  general  wave  repre- 
sented by  the  equivalent  sine  wave,  that  is,  the  sine  wave  of  the 
same  effective  value  as  the  general  wave. 


POLAR  COORDINATES  AND  POLAR  DIAGRAMS    53 


The  transformation  of  the  general  wave  into  the  equivalent 
sine  wave,  however,  has  to  be  carried  out  algebraically  in  the 
crank  diagram,  while  the  polar  diagram  permits  a  graphical 
transformation  of  the  general  wave  into  the  equivalent  sine  wave. 

Let  Fig.  48  represent  a  general  alternating  wave.  An  element 
BiOB2  of  this  wave  then  has  the  area 

dA  =  r-Y> 

and    the    total   area   of   the   polar 
curve  is 


-  n 

Jo     : 


A  = 

The  effective  value  of  the  wave  is 
R      =  \/mean  square 


hence, 


FIG.  48. 


R< 


-if 


r*de  =  A. 


The  area  of  the  polar  curve  of  the  general  periodic  wave,  as 
measured  by  planimeter,  therefore  equals  the  area  of  a  circle 
with  the  effective  value  of  the  wave  as  radius. 

The  effective  value  of  the  equivalent  sine  wave  therefore  is 
the  radius  of  a  circle  having  the  same  area  as  the  general  wave, 
in  polar  coordinates: 

II 


R  =  A  - 


The  diameter  of  the  general  polar  circle,  therefore,  is 


And  the  phase  of  the  equivalent  sine  wave,  or  the  direction  of 
the  diameter  of  its  polar  circle,  is  the  vector  bisecting  the  area 
of  the  general  wave,  in  polar  coordinates. 

The  transformation  of  the  general  alternating  wave  into  the 
equivalent  sine  wave,  therefore,  is  carried  out  by  measuring  the 
area  of  the  general  wave  in  polar  coordinates,  and  drawing  the 
sine  wave  circle  of  half  this  area. 


SECTION  II 

CIRCUITS 

CHAPTER  VIII 
ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE 

48.  If  in  a  continuous-current  circuit,  a  number  of  resistances, 
Tit  rz>  rs>  •  •  •>  are  connected  in  series,  their  joint  resistance,  R, 
is  the  sum  of  the  individual  resistances,  R  =  ri  +  ?*2  +  rs  +  .  .  . 

If,  however,  a  number  of  resistances  are  connected  in  multiple 
or  in  parallel,  their  joint  resistance,  R,  cannot  be  expressed  in  a 
simple  form,  but  is  represented  by  the  expression 

R  =  ~z i ^ • 


Hence,  in  the  latter  case  it  is  preferable  to  introduce,  instead  of 
the  term  resistance,  its  reciprocal,  or  inverse  value,  the  term 

conductance,  g  =  -•        If,    then,    a    number    of    conductances, 

9i>  92>  03>  •  •  •  are  connected  in  parallel,  their  joint  conductance 
is  the  sum  of  the  individual  conductances,  or  G  =  gi  +  gr2  + 
03  -h  .  .  .  When  using  the  term  conductance,  the  joint  con- 
ductance of  a  number  of  series-connected  conductances  becomes 
similarly  a  complicated  expression 


.. 

01  02  03 

Hence  the  term  resistance  is  preferable  in  case  of  series  con- 
nection, and  the  use  of  the  reciprocal  term  conductance  in  parallel 
connections  ;  therefore, 

The  joint  resistance  of  a  number  of  series-connected  resistances 
is  equal  to  the  sum  of  the  individual  resistances;  the  joint  conduct- 
ance of  a  number  of  parallel-connected  conductances  is  equal  to 
the  sum  of  the  individual  conductances. 

54 


ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE  55 

49.  In  alternating-current  circuits,  instead  of  the  term  resist- 
ance we  have  the  term  impedance,  Z  =  r  +  jx,  with  its  two 
components,  the  resistance,  r,  and  the  reactance,  x,  in  the  formula 
of  Ohm's  law,  E  =  IZ.  The  resistance,  r,  gives  the  component 
of  e.m.f.  in  phase  with  the  current,  or  the  power  component 
of  the  e.m.f.,  Ir;  the  reactance,  x,  gives  the  component  of  the 
e.m.f.  in  quadrature  with  the  current,  or  the  wattless  component 
of  e.m.f.,  Ix;  both  combined  give  the  total  e.m.f., 

Iz  =  iVr2  +  z2. 

Since  e.m.fs.  are  combined  by  adding  their  complex  expressions, 
we  have: 

The  joint  impedance  of  a  number  of  series-connected  impedances 
is  the  sum  of  the  individual  impedances,  when  expressed  in  com- 
plex quantities. 

In  graphical  representation  impedances  have  not  to  be  added, 
but  are  combined  in  their  proper  phase  by  the  law  of  parallelo- 
gram in  the  same  manner  as  the  e.m.fs.  corresponding  to  them. 

The  term  impedance  becomes  inconvenient,  however,  when 
dealing  with  parallel-connected  circuits;  or,  in  other  words,  when 
several  currents  are  produced  .by  the  same  e.m.f.,  such  as  in 
cases  where  Ohm's  law  is  expressed  in  the  form, 

i«| 

.       Z 

It  is  preferable,  then,  to  introduce  the  reciprocal  of  impe- 
dance,  which  may  be  called  the  admittance  of  the  circuit,  or 


As  the  reciprocal  of  the  complex  quantity,  Z  =  r  +  j%,  the 
admittance  is  a  complex  quantity  also,  or  Y  =  g  —  jb;  it  con- 
sists of  the  component,  g,  which  respresents  the  coefficient  of 
current  in  phase  with  the  e.m.f.,  or  the  power  or  active  com- 
ponent, gE,  of  the  current,  in  the  equation  of  Ohm's  law, 

I  =YE  =  (g-jb)E, 

and  the  component,  b,  which  represents  the  coefficient  of  current 
in  quadrature  with  the  e.m.f.,  or  wattless  or  reactive  component, 
bE,  of  the  current. 

g  is  called  the  conductance,  and  b  the  susceptance,  of  the  cir- 
cuit.    Hence  the  conductance,  g,  is  the  power  component,  and 


56  ALTERNATING-CURRENT  PHENOMENA 

the  susceptance,  b,  the  wattless  component,  of  the  admittance, 
Y  =  g  —  jb,  while  the  numerical  value  of  admittance  is 

the  resistance,  r,  is  the  power  component,  and  the  reactance, 
x,  the  wattless  component,  of  the  impedance,  Z  =  r  +  jx,  the 
numerical  value  of  impedance  being 

z  =  ->/r2  +  x2. 

50.  As  shown,  the  term  admittance  implies  resolving  the  cur- 
rent into  two  components,  in  phase  and  in  quadrature  with  the 
e.m.f.,  or  the  power  or  active  component  and  the  wattless  or 
reactive  component;  while  the  term  impedance  implies  resolving 
the  e.m.f.  into  two  components,  in  phase  and  in  quadrature 
with  the  current,  or  the  power  component  and  the  wattless  or 
reactive  component. 

It  must  be  understood,  however,  that  the  conductance  is  not 
the  reciprocal  of  the  resistance,  but  depends  upon  the  reactance 
as  well  as  upon  the  resistance.  Only  when  the  reactance  x  =  0, 
or  in  continuous-current-circuits,  is  the  conductance  the  recip- 
rocal of  resistance. 

Again,  only  in  circuits  with  zero  resistance  (r  —  0)  is  the 
susceptance  the  reciprocal  of  reactance;  otherwise,  the  suscep- 
tance depends  upon  reactance  and  upon  resistance. 

The  conductance  is  zero  for  two  values  of  the  resistance: 

1.  Ifr=  oo?  or  z  =  oo,  since  in  this  case  there  is  no  current, 
and  either  component  of  the  current  =  0. 

2.  If  r  =  0,  since  in  this  case  the  current  in  the  circuit  is  in 
quadrature  with  the  e.m.f.,  and  thus  has  no  power  component. 

Similarly,  the  susceptance,  6,  is  zero  for  two  values  of  the 
reactance: 

1.  If  x  =  c° ,  or  r  =  oo. 

2.  If  x  =  0. 

From  the  definition  of  admittance,  Y  =  g  —  jb,  as  the  recip- 
rocal of  the  impedance,  Z  =  r  +  jx, 
we  have 

1  * 

Y  =  7^,  or  g  —  jb  =  ^   .    .^ 

or,  multiplying  numerator  and  denominator  on  the  right  side  by 

(r-jx), 

r  -  jx 


ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE  57 

hence,  since 

(r  +  jx)  (r  -  jx)  =  r2  +  x2  =  z2, 

r  .       x  r        .  x 

*          x2    -  3  r2  +  x2    =    ~*   ~  3  ~2 


or 


L 

z2 

X  X 


9  ~  r2  -f  x2       z2' 


(/      ~~~  O         I  O       ~~  *>  J 

r2  +  x2       z2 
and  conversely 

.  £ 

y»' 

A 

ya- 

By  these  equations,  the  conductance  and  susceptance  can  be 
calculated  from  resistance  and  reactance,  and  conversely. 
Multiplying  the  equations  for  g  and  r,  we  get 


hence,  z2y2  =  (r2  +  x2)  (g2  +  b2)  =  1; 

1  1  ]       the  absolute  value 

and  2  =  - 


+  b2  \       of  impedance; 

1  1  I       the  absolute  value 


z       -\/r2  _j_  X2  j       Of  admittance. 

51.  If,  in  a  circuit,  the  reactance,  x,  is  constant,  and  the 
resistance,  r,  is  varied  from  r  =  0  to  r  —  <» ,  the  susceptance, 

b,  decreases  from  6  =  -  at  r  =  0,  to  6  =  0  at  r  =  °°  ;  while  the 

M/ 

conductance,  g  =  0  at  r  =  0,  increases,  reaches  a  maximum  for 
T*  =  x,  where  g  =  ~— ,  is  equal  to  the  susceptance  or  g  =  b,  and 

then  decreases  again,  reaching  g  =  0  at  r  =  °° . 

In  Fig.  49,  for  constant  reactance  x  =  0.5  ohm,  the  variation 
of  the  conductance,  g,  and  of  the  susceptance,  6,  are  shown  as 
functions  of  the  varying  resistance,  r.  As  shown,  the  absolute 
value  of  admittance,  susceptance,  and  conductance  are  plotted 
in  full  lines,  and  in  dotted  line  the  absolute  value  of  impedance, 

z  = 


58 


ALTERNATING-CURRENT  PHENOMENA 


Obviously,  if  the  resistance,  r,  is  constant,  and  the  reactance, 
x,  is  varied,  the  values  of  conductance  and  susceptance  are 
merely  exchanged,  the  conductance  decreasing  steadily  from 

g  =  -  to  0,  and  the  susceptance  passing  from  0  at  x  =  0  to  the 


maxmum,      =  x-  =  gf  =  ^— 

_  /'  z  x 


,  and  to  6  =  0  at  x  = 


The  resistance,  r,  and  the  reactance,  x,  vary  as  functions  of 
the  conductance,  g,  and  the  susceptance,  b,  in  the  same  manner 
as  g  and  6  vary  as  functions  of  r  and  x. 


OHlJs 

n  n\- 

3 

\ 

L8 

1.7 
1.6 
LS 
1.4 

u 

1.2 

U 

1.0 

.« 

4 

.7 

4 

.5 

Y 

\ 

RE; 

CT; 

NC 

CO 

NS1 

ANT 

=  .E 

OH 

MS 

s 

\ 

\ 

^ 

\ 

\ 

/ 

\ 

s 

/ 

s 

\ 

\ 

s 

/ 

\ 

1 

i 

'£ 

\ 

'^ 

\ 

^>1 

f? 

\ 

* 

\ 

'$ 

\ 

^>S 

\/ 

/ 

/ 

c 

^^ 

^H.' 

XX 

X 

/ 

\ 

.  / 

[ 

^v 

^x, 

^ 

( 

/ 

k, 

^ 

"^^ 

^ 

/ 

s 

'' 

\ 

^S 

~^~ 

<^ 

\ 

•  —  ^ 

J 

^^ 

5 

bi 

^5 

^ 

.4 

1$ 

w 

N 

/ 

y 

X 

^x, 

2 

^^ 

-^, 

*-*.^^ 

.1 
° 

1 

*****. 

•—  -^ 

•~-~. 

/  1 

R 

SISTAN 

DE: 

',  Oh 

MS 

>     .1     £     ,8    ^     ,6     J&     .7     .8     .9     1.0  1.1  1.2    1.3   1.4  1.- 

FIG.  49. 

1,0  1.7  1.8 

The  sign  in  the  complex  expression  of  admittance  is  always 
opposite  to  that  of  impedance;  this  is  obvious,  since  if  the  cur- 
rent lags  behind  the  e.m.f.,  the  e.m.f.  leads  the  current,  and 
conversely. 

We  can  thus  express  Ohm's  law  in  the  two  forms, 

E  =  IZ,~ 
I  =EY, 

and  therefore, 


ADMITTANCE,  CONDUCTANCE,  SUSCEPTANCE  59 

The  joint  impedance  of  a  number  of  series-connected  impedances 
is  equal  to  the  sum  of  the  individual  impedances;  the  joint  admit- 
tance of  a  number  of  parallel-connected  admittances  is  equal  to  the 
sum  of  the  individual  admittances,  if  expressed  in  complex  quantities. 
In  diagrammatic  representation,  combination  by  the  parallelogram 
law  takes  the  place  of  addition  of  the  complex  quantities. 

62.  Experimentally,  impedances  and  admittances  are  most 
conveniently  determined  by  establishing  an  alternating  current 
in  the  circuit,  and  measuring  by  voltmeter,  ammeter  and  watt- 
meter, the  volts,  e,  the  amperes,  i,  and  the  watts,  p. 

It  is  then, 

Impedance:  z  =  -• 

P 

Resistance  (effective):  r  =  ^ 

Reactance:  x  = 

f\ 

Admittance:  y  =  — 

6 

Conductance:  g  =  -$• 
Susceptance:  b  =  \/y2  —  g2. 

Regarding  their  calculation,  see  "Theoretical  Elements  of 
Electrical  Engineering." 


CHAPTER  IX 

CIRCUITS    CONTAINING    RESISTANCE,    INDUCTIVE 
REACTANCE,  AND  CONDENSIVE  REACTANCE 

53.  Having,  in  the  foregoing,  re-established  Ohm's   law  and 
KirchhofTs  laws  as  being  also  the  fundamental  laws  of  alternating- 
current  circuits,  when  expressed  in  their  complex  form, 

E  =  ZI,  or,  7  =  YE, 

and 

2E  =  0  in  a  closed  circuit, 

2  7   =  0  at  a  distributing  point t 

where  E,  I,  Z,  Y,  are  the  expressions  of  e.m.f.,  current,  impe- 
dance, and  admittance  in  complex  quantities — these  values 
representing  not  only  the  intensity,  but  also  the  phase,  of  the 
alternating  wave — we  can  now — by  application  of  these  laws, 
and  in  the  same  manner  as  with  continuous-current  circuits, 
keeping  in  mind,  however,  that  E,  I,  Z,  Y,  are  complex  quanti- 
ties— calculate  alternating-current  circuits  and  networks  of 
circuits  containing  resistance,  inductive  reactance,  and  conden- 
sive  reactance  in  any  combination,  without  meeting  with  greater 
difficulties  than  when  dealing  with  continuous-current  circuits. 
It  is  obviously  not  possible  to  discuss  with  any  completeness 
all  the  infinite  varieties  of  combinations  of  resistance,  inductive 
reactance,  and  condensive  reactance  which  can  be  imagined, 
and  which  may  exist,  in  a  system  of  network  of  circuits;  there- 
fore only  some  of  the  more  common  or  more  interesting  combina- 
tions will  here  be  considered. 

1.  Resistance  in  Series  with  a  Circuit 

54.  In   a   constant-potential   system   with   impressed   e.m.f., 

Eo  =  eQ  +  je'o,  EQ  =  Ve<>2  +  e0'2, 

let  the  receiving  circuit  of  impedance, 

Z  =  r  +  jx,  z  =  Vr2  +  x2, 

be  connected  in  series  with  a  resistance,  r0. 

60 


CIRCUITS  CONTAINING  RESISTANCE  61 

The  total  impedance  of  the  circuit  is  then 
Z  +  rQ  =  r  +  r0  +  jx\ 
hence  the  current  is 

ffo  #o  #o(r  +  r0  -  jap 

''    (r  +  r0)2  +  z2  ; 


and  the  e.m.f.  of  the  receiving  circuit  becomes 
F        17  = 


z2  +  2  rr0  +  ro2     ' 

or,  in  absolute  values  we  have  the  following: 
Impressed  e.m.f., 


current, 

,  _  EQ  EQ 


V(r  +  r0)2  -1-  z2      Vz2  +  2  rr0  +  r02 ' 
e.m.f.  at  terminals  of  receiver  circuit, 


(r  +  r0)2  +  a;2 
difference  of  phase  in  receiver  circuit,  tan  6  =  -; 

difference  of  phase  in  supply  circuit,  tan  00  = 


r0 
since  in  general, 

x      imaginary  component 

tan  (phase)  =  -  -7 

real  component 

(a)  If  x  is  negligible  with  respect  to  r,  as  in  a  non-inductive 
receiving  circuit, 

T  -        E°  F  -     W         T 

J-     —          j          )  Hi    —    -C/o         i          > 

r  +  ro  r  +  r0 

and  the  current  and  e.m.f.  at  receiver  terminals  decrease  steadily 
with  increasing  r0. 

(6)  If  r  is  negligible  compared  with  x,  as  in  a  wattless  receiver 
circuit, 

J-          Eo—         F       F  X 

J.     —  / ,         Jli     —    JG/n          , 

Vro2  +  z2  Vro2  +  a:2' 

or,  for  small  values  of  r0, 

/    =   —  >  E   =   EQ', 

JO 


62 


ALTERNATING-CURRENT  PHENOMENA 


that  is,  the  current  and  e.m.f.  at  receiver  terminals  remain 
approximately  constant  for  small  values  of  r0,  and  then  de- 
crease with  increasing  rapidity. 

In  the  general  equations,  x  appears  in  the  expressions  for 
/  and  E  only  as  x2,  so  that  /  and  E  assume  the  same  value  when 
x  is  negative  as  when  x  is  positive;  or,  in  other  words,  series 
resistance  acts  upon  a  circuit  with  leading  current,  or  in  a 
condenser  circuit,  in  the  same  way  as  upon  a  circuit  with  lag- 
ging current,  or  an  inductive  circuit. 

For  a  given  impedance,  2,  of  the  receiver  circuit,  the  current, 
7,  and  e.m.f.,  E,  are  smaller  the  larger  the  value  of  r;  that  is, 
the  less  the  difference  of  phase  in  the  receiver  circuit. 


100 


060 


IMPRESSED  E..M.F.  CONSTANT,  E0=IOO 
IMPEDANCE  OF  RECEIVER  CIRCUIT  CONSTANT,  Z  «  f.O 
LINE-  RESISTANCE  CONSTANT 


DUdlTAf. 


REACTANCE 


-hi   T.2--.3--.4  --.S--.6  •- 


OONDEN3AN 


FIG.  50. — Variation  of  voltage  at  constant  series  resistance   with  phase 
relation  of  receiver  circuit. 

As  an  instance,  in  Fig.  50  is  shown  the  e.m.f.,  E,  at  the  re- 
ceiver circuit,  for  EQ  =  const.  =  100  volts,  z  =  1  ohm;  hence 
I  =*  E,  and 

(a)  r0  =  0.2  ohm         (Curve    I) 
(6)  r0  =  0.8  ohm         (Curve  II) 

for   abscissae,    from 


with   values   of   reactance,    x  = 
x  =  +  1.0  to  x  =  -  1.0  ohm. 

As  shown,  /  and  E  are  smallest  for  x  =  0,  r  =  1.0,  or  for 
the  non-inductive  receiver  circuit,  and  largest  for  x  =  ±1.0, 
r  =  0,  or  for  the  wattless  circuit,  in  which  latter  a  series  resist- 
ance causes  but  a  very  small  drop  of  potential. 

Hence  the  control  of  a  circuit  by  series  resistance  depends 
upon  the  difference  of  phase  in  the  circuit. 


CIRCUITS  CONTAINING  RESISTANCE 


63 


For  r0  =  0.8  and  x  =  0,  x  =  +  0.8,  x  =  —  0.8,  the  vector 
diagrams  are  shown  in  Figs.  51  to  53. 

In  these  Figs.  OEQ  is  the  supply  voltage,  OES  the  voltage  con- 
sumed by  the  line  resistance,  and  OE  thej^eceiver  voltage,  with 
its  two  components,  OEi  in  phase  and  OE2  in  quadrature  with 
the  current. 


Es  E 


FIG.  52. 


FIG.  53. 


FIG.  51. 

2.  Reactance  in  Series  with  a  Circuit 

52.  In    a    constant    potential    system    of    impressed    e.m.f., 
EQ  =  e»  +  je'o,     E0  =  Ve'o2  +  e'02 

let  a  reactance,  x0,  be  connected  in  series  in  a  receiver  circuit  of 
impedance, 

Z  =  r  +  jx,        z  =  Vr2  +  z2. 

Then,  the  total  impedance  of  the  circuit  is 

Z  +  jxQ  =  r  +  j  (x  +  £0), 
and  the  current  is 


"  Z  +  JXQ  ~  r  +  j  (x  +  zo) 
while  the  difference  of  potential  at  the  receiver  terminals  is 


E  =  IZ  =  E0 

•        • 

Or,  in  absolute  quantities, 
current, 

E0 


+  jx 


r+j(x 


I  = 


E. 


(x 


e.m.f.  at  receiver  terminals, 


2xxo 
E0z 


64  ALTERNATING-CURRENT  PHENOMENA 

difference  of  phase  in  receiver  circuit, 

x 
tan  6  =  -; 

difference  of  phase  in  supply  circuit, 

tan  e«  =  - 


(a)  If  x  is  small  compared  with  r,  that  is,  if  the  receiver  circuit 
is  non-inductive,  /  and  E  change  very  little  for  small  values  of 
XQ',  but  if  x  is  large,  that  is,  if  the  receiver  circuit  is  of  large  re- 
actance, 7  and  E  change  considerably  with  a  change  of  x0. 

(b)  If  x  is  negative,  that  is,  if  the  receiver  circuit  contains 
condensers,    synchronous    motors,    or    other    apparatus    which 
produce  leading  currents,  below  a  certain  value  of  XQ  the  de- 
nominator in  the  expression  of  E  becomes  <z,  or  E  >  EQ;  that 
is,  the  reactance,  x0,  raises  the  voltage. 

(c)  E  =  EQ)   or  the  insertion  of  a  series  reactance,  XQ,  does 
not  affect  the  potential  difference  at  the  receiver  terminals,  if 


2x  XQ  -f  x02  =  z; 
or,  XQ  =  —  2x. 

That  is,  if  the  reactance  which  is  connected  in  series  in  the 
circuit  is  of  opposite  sign,  but  twice  as  large  as  the  reactance 
of  the  receiver  circuit,  the  voltage  is  not  affected,  but  E  =  EQ, 

ET 

/  =  ~    If  xQ  <  —  2x,  it  raises,  if  XQ  >  —  2x,  it  lowers,  the  voltage. 

We  see,  then,  that  a  reactance  inserted  in  series  in  an  alter- 
nating-current circuit  will  always  lower  the  voltage  at  the 
receiver  terminals,  when  of  the  same  sign  as  the  reactance  of  the 
receiver  circuit;  when  of  opposite  sign,  it  will  lower  the  voltage 
if  larger,  raise  the  voltage  if  less,  than  twice  the  numerical  value 
of  the  reactance  of  the  receiver  circuit. 

(d)  If  x  =  0,  that  is,  if  the  receiver  circuit  is  non-inductive, 
the  e.m.f.  at  receiver  terminals  is 

" 


E  = 

' 


(     /         =  =  (1  -|-  x)      ^  expanded  by  the  binomial  theorem 

n  i  n(n  ~  ^    21         \ 

~^T~X    '••')' 


CIRCUITS  CONTAINING  RESISTANCE 
Therefore,  if  x0  is  small  compared  with  r, 


65 


-  E 


That  is,  the  percentage  drop  of  potential  by  the  insertion 
of  reactance  in  series  in  a  non-inductive  circuit  is,  for  small 
values  of  reactance,  independent  of  the  sign,  but  proportional 
to  the  square  of  the  reactance,  or  the  same  whether  it  be  induc- 
tive reactance  or  condensive  reactance.  ,- 

56.  As  an  example,  in  Fig.  54  the  changes  of  current,  7,  and 
of  e.m.f.  at  receiver  terminals,  E}  at  constant  impressed  e.m.f., 

VOLTS  E  OR  AMPERES  I 


100 
90 

»80 

j 
c  70 

IMPRESSED  E.M.F.  CON 
IMPEDENCE  OF  RECEIVER  Clf 
I    r=1.0  x=0           "5- 

HI  r  =  -6    s-=-'.8 

5TANT, 
?CUIT  < 
1.0  f 

-E°N 

100 
ST 

^NT 

11 
16 

9  L 

0 

/- 

^N 

/ 

\ 

15 

l> 

/ 

\ 

/ 

\ 

14 

0 

/ 

\ 

1 

\ 

13 

0 

/ 

\ 

/ 

\ 

12 

0 

/ 

\ 

/ 

\H 

o/ 

/ 

\ 

/ 

i 

i/ 

^ 

^ 

A 

^ 

\ 

s 

/ 

7 

/ 

/ 

8 

\ 

0 

\ 

s 

\ 

/ 

7 

/ 

/ 

/ 

7 

0 

\ 

\ 

\ 

j'° 

L.    gQ 

/ 

* 

\, 

/ 

^ 

/ 

6 

0 

> 

SN 

\ 

. 

Ern 

/ 

^ 

^ 

a 

^ 

5 

0 

X 

c  oU 
>40 

J    OA 

^ 

^ 

^ 

^ 

-^ 

4 

0 

^ 

,  

—^- 

~~~~ 

^~-  -—  • 

—  •—  • 

— 

3 

0 

5U° 

2 

0 

} 

8. 

a?0-f  3.0  2.8  2.6  2.4  2.2  2.0  1.8  1.6  1.4  1.2  1.0  .8   .6    .4  +.2     0  -.2   .4     .6    .8    1.0  1.2-1.4 
OHMS  INDUCTANCE-*—  REACTANCE-*- CONDENSANCE 

FlG.    54. 

EQ,  are  shown  for  various  conditions  of  a  receiver  circuit  and 

amounts  of  reactance  inserted  in  series. 

Fig.  54  gives  for  various  values  of  reactance,  XQ  (if  positive, 

inductive;  if  negative,  condensive),  the  e.m.fs.,  E,  at  receiver 

terminals,  for  constant  impressed  e.m.f.,  EQ  =  100  volts,  and 

the  following  conditions  of  receiver  circuit: 
z  =  1.0,  r  =  1.0,  x  =  0  (Curve  I) 
z  =  i.o,  r  =  0.6,  x  =  0.8  (Curve  II) 
z  =  i.o,  r  =  0.6,  x  =  -  0.8  (Curve  III). 


66 


ALTERNATING-CURRENT  PHENOMENA 


As  seen,  curve  I  is  symmetrical,  and  with  increasing  z0  the 
voltage  E  remains  first  almost  constant,  and  then  drops  off 
with  increasing  rapidity. 

In  the  inductive  circuit  series  inductive  reactance,  or  in  a 
condenser  circuit  series  condensive  reactance,  causes  the  voltage 
to  drop  off  very  much  faster  than  in  a  non-inductive  circuit. 

Series  inductive  reactance  in  a  condenser  circuit,  and  series 
condensive  reactance  in  an  inductive  circuit,  cause  a  rise  of 
potential.  This  rise  is  a  maximum  for  x0  =  ±  0.8,  or  XQ  = 
—  x  (the  condition  of  resonance),  and  the  e.m.f.  reaches  the 

value  E  =  167   volts,   or  E  —  E0-'    This  rise  of  potential  by 

series  reactance  continues  up  to  XQ  =  ±  1.6,  or,  XQ  =  —  2x, 
where  E  =  100  volts  again;  and  for  XQ  >  1.6  the  voltage  drops 
again. 

At  x0  =  ±  0.8,  x  =  +  0.8,  the  total  impedance  of  the  circuit 
is  r  —  j  (x  +  XQ)  =  r  =  0.6,  x  +  XQ  =  0,  and  tan  00  =  0;  that 


FIG.  55. 


FIG.  56. 


FIG.  57. 


is,  the  current  and  e.m.f.  in  the  supply  circuit  are  in  phase  with 
each  other,  or  the  circuit  is  in  electrical  resonance. 

Since  a  synchronous  motor  in  the  condition  of  efficient  work- 
ing acts  as  a  condensive  reactance,  we  get  the  remarkable  result 
that,  in  synchronous  motor  circuits,  choking  coils,  or  reactive 
coils,  can  be  used  for  raising  the  voltage. 

In  Figs.  55  to  57,  the  vector  diagrams  are  shown  for  the 
conditions 


=  100,  XQ  =  0.6,  x  =  0 
x  =  +  0.8 
x  =  -  0.8 


(Fig.  48)  E  =  85.7 
(Fig.  49)  E  =  65.7 
(Fig.  50)  E  =  158.1. 


CIRCUITS  CONTAINING  RESISTANCE 


67 


57.  In  Fig.  58  the  dependence  of  the  potential,  E,  upon  the 
difference  of  phase,  6,  in  the  receiver  circuit  is  shown  for  the 
constant  impressed  e.m.f.,  EQ  =  100;  for  the  constant  receiver 
impedance,  z  =  1.0  (but  of  various  phase  differences  0),  and  for 
various  series  reactances,  as  follows: 

x0  =  0.2         (Curve  I) 

x0  =  0.6         (Curve  II) 

x0  =  0.8         (Curve  III) 

XQ  =  1.0         (Curve  IV) 

BO  =  1.6         (Curve  V) 

x0  =  3.2         (Curve  VI). 

Since  z  =  1.0,  the  current,  /,  in  all  these  diagrams  has  the 
same  value  as  E. 


180 
170 
160 
150 
140 
130 


IMPRESSED  E.M.F,  CONSTANT,  E0=100 
IMPEDANCE  OF  RECEIVER  CIRCUIT  CONSTANT, 

I,  *o-.2  IV,  x  o=1.0  Z=1.0 

[I,  ff0  =  .6  V.    «0=1.6 

VI,   *o=3.2  y 


/// 


80  70  60  50  40  30  20  10  0  10  20  30  40  50  60  70  80  90  DEGREES 

LAG^-PHASE  DIFFERENCE  IN  CONSUMER  CJRCUIT-»-LEAD 
FIG.  58. 

In  Figs.  59  and  60,  the  same  curves  are  plotted  as  in  Fig.  58, 
but  in  Fig.  59  with  the  reactance,  x,  of  the  receiver  circuit  as 
abscissas;  and  in  Fig.  60  with  the  resistance,  r,  of  the  receiver 
circuit  as  abscissas. 

As  shown,  the  receiver  voltage,  E}  is  always  lowest  when  XQ 
and  x  are  of  the  same  sign,  and  highest  when  they  are  of  opposite 
sign. 


68 


ALTERNA  TING-CURRENT-PHENOMENA 


IMPRESSED  E.M.F.  CONSTANT,   E0  =  100 
IMPEDANCE  O.F  RECEIVER  CIRCUIT  CONSTANT.  Z«1.0 


10 


•f  1  +.9  +.8  +.7  +.6  +.5  +.4  +.3  4.2  +.1  0      -.1   -.2  -.3  -.4  -.5  -.6   -.7  -.8  -.9-10 
REACTANCE  OF  CONSUMER  CIRCUIT 

FIG.  59. 


IMPRESSED  E.M.F,  CONSTANT,  E0  =  100 
IMPEDANCE  OF  RECEIVER  CIRCUIT 
CONSTANT.  Z  -1.0 


0    .1    .2     .3    .4     .5     .6     .7 
LAGGING  CURRENT   -« 


_     1.0  .9     , 

ESISTANCE  OF 
CONSUMER  CIRCUIT 

FIG.  60. 


8    .7    .6 


.5    .4    .3    .2    .1   .0 
LEADING  CURRENT 


CIRCUITS  CONTAINING  RESISTANCE  69 

The  rise  of  voltage  due  to  the  balance  of  XQ  and  x  is  a  maxi- 
mum for  XQ  =  +  1.0,  x  =  —  1.0,  and  r  =  0,  where  E  =  oo  ; 
that  is,  absolute  resonance  takes  place.  Obviously,  this  condi- 
tion cannot  be  completely  reached  in  practice. 

It  is  interesting  to  note,  from  Fig.  60,  that  the  largest  part 
of  the  drop  of  potential  due  to  inductive  reactance,  and  rise  to 
condensive  reactance — or  conversely — takes  place  between 
r  =  1.0  and  r  =  0.9;  or,  in  other  words,  a  circuit  having  a 
power-factor  cos  6  =  0.9  gives  a  drop  several  times  larger  than 
a  non-inductive  circuit,  and  hence  must  be  considered  as  an 
inductive  circuit. 

3.  Impedance  in  Series  with  a  Circuit 

68.  By  the  use  of  reactance  for  controlling  electric  circuits, 
a  certain  amount  of  resistance  is  also  introduced,  due  to  the 
ohmic  resistance  of  the  conductor  and  the  hysteretic  loss,  which, 
as  will  be  seen  hereafter,  can  be  represented  as  an  effective 
resistance. 

Hence  the  impedance  of  a  reactive  coil  (choking  coil)  may  be 
written  thus: 


Z0  =  r0  +  jxQ,  ZQ  = 

where  r0  is  in  general  small  compared  with 
From  this,  if  the  impressed  e.m.f.  is 


E  o  =  eQ  +  je'o,  EQ  =  VV  +  *o' 

and  the  impedance  of  the  consumer  circuit  is 

Z  =  r  +  jx,  z  =  \/V2  -f  x2, 

we  get  the  current 

,    _  EQ          __  EQ  _ 

~  Z  +  ZQ~  (r  +  r0)  +  j(z  +  z0) 
and  the  e.m.f.  at  receiver  terminals, 

Z  r+x 


Or,  in  absolute  quantities, 
the  current  is, 

E. 


I  = 


V(r  +  r0)2  -f  (z  +  Z0)2       V  z2  +  z02  +  2  (rr0  +  zz0) 
the  e.m.f.  at  receiver  terminals  is 

E  = 


(x  +  z0)2  z2  +  zo2  +  2  (rr 


70          ALTERNATING-CURRENT  PHENOMENA 

the  difference  of  phase  in  receiver  circuit  is 

or 

tan  0  —  — ; 

and  the  difference  of  phase  in  the  supply  circuit  is 

tan  0  =  — ; — -• 

f   T"  7*0 

69.  In  this  case,  the  maximum  drop  of  potential  will  not 
take  place  for  either  x  =  0,  as  for  resistance  in  series,  or  for 
r  =  0,  as  for  reactance  in  series,  but  at  an  intermediate  point. 
The  drop  of  voltage  is  a  maximum;  that  is,  E  is  a  minimum  if 
the  denominator  of  E  is  a  maximum;  or,  since  z,  z0,  r0,  x0  are 
constant,  if  rr0  +  XXQ  is  a  maximum,  that  is,  since  x  =  \/zz  —  r2, 
if  rr0  -\-  xQ\/z2  —  r2  is  a  maximum.  A  function,  /  =  rr0  +  XQ 
Vz2  —  r2,  is  a  maximum  when  its  differential  coefficient  equals 
zero.  For,  plotting  /  as  curve  with  values  of  r  as  abscissas,  at 
the  point  where  /  is  a  maximum  or  a  minimum,  this  curve  is 
for  a  short  distance  horizontal,  hence  the  tangens-function  of 
its  tangent  equals  zero.  The  tangens-function  of  the  tangent 
of  a  curve,  however,  is  the  ratio  of  the  change  of  ordinates  to 
the  change  of  abscissas,  or  is  the  differential  coefficient  of  the 
function  represented  by  the  curve. 

Thus  we  have 

/  =  rr0  +  xQ\/z2  —  r2 

is  a  maximum  or  minimum,  if 
Differentiating,  we  get 

1  /y- 

-  2r)  =  0; 

=  0, 

That  is,  the  drop  of  potential  is  a  maximum,  if  the  reactance 
factor,  -,  of  the  receiver  circuit  equals  the  reactance  factor,  — °, 

T  TQ 

of  the  series  impedance. 

60.  As  an  example,  Fig.  61  shows  the  e.m.f.,  E,  at  the  receiver 
terminals,  at  a  constant  impressed  e.m.f.,  EQ  =  100,  a  constant 


CIRCUITS  CONTAINING  RESISTANCE 


71 


impedance  of  the  receiver  circuit,  z  =  1.0,  and  constant  series 

impedances, 

Z0  =  0.3  -h  j  0.4  (Curve  I) 

Z0  =  1.2 +j  1.6  (Curve  II) 

as  functions  of  the  reactance,  x,  of  the  receiver  circuit. 


150 
140 


130 
120 


110 
100 


90 


70 


50 


1.  .9  .8  .7  .6  ,5  A  .3  ..2  .1  0  -.1  -,2  -,3  -,4  -.5  -.6  -J  -.8  -.9-1, 


FlG.  62. 


FIG.  63. 


72 


ALTERNATING-CURRENT  PHENOMENA 


Figs.  62  to  64,  give  the  vector  diagram  for  EQ  =  100,  x  =  0.95, 
x  =  o,  x  =  -  0.95,  and  ZQ  =  0.3  +  0.4  j. 


4.  Compensation  for  Lagging  Currents  by  Shunted 
Condensive  Reactance 

61.  We  have  seen  in  the  preceding  paragraphs,  that  in  a 
constant  potential  alternating-current  system,  the  voltage  at 
the  terminals  of  a  receiver  circuit  can  be  varied  by  the  use  of 
a  variable  reactance  in  series  with  the  circuit,  without  loss  of 
energy  except  the  unavoidable  loss  due  to  the  resistance  and 
hysteresis  of  the  reactance;  and  that,  if  the  series  reactance  is 
very  large  compared  with  the  resistance  of  the  receiver  circuit, 
the  current  in  the  receiver  circuit  becomes  more  or  less  inde- 
pendent of  the  resistance — that  is,  of  the  power  consumed  in 
the  receiver  circuit,  which  in  this  case  approaches  the  conditions 
of  a  -constant  alternating-current  circuit,  whose  current  is 


I  = 


E 
-,  or,  approximately,  I  =  — • 


This  potential  control,  however,  causes  the  current  taken 
from  the  mains  to  lag  greatly  behind  the  e.m.f.,  and  thereby 
requires  a  much  larger  current  than  corresponds  to  the  power 
consumed  in  the  receiver  circuit. 

Since  a  condenser  draws  from  the  mains  a  current  in  leading 
phase,  a  condenser  shunted  across  such  a  circuit  carrying  cur- 
rent in  lagging  phase  compensates  for  the  lag,  the  leading  and 
the  lagging  current  combining  to  form  a  resultant  current  more 


CIRCUITS  CONTAINING  RESISTANCE 


73 


or  less  in  phase  with  the  e.m.f.,  and  therefore  proportional  to 
the  power  expended. 

In  a  circuit  shown  diagrammatically  in  Fig.  65,  let  the  non- 
inductive  receiver  circuit  of  resistance,  r,  be  connected  in  series 
with  the  inductive  reactance,  XQ)  and  the  whole  shunted  by  a 
condenser  C  of  condensive  reactance,  xe,  entailing  but  a  negligible 
loss  of  power. 


FIG.  65. 

Then,  if  EQ  =  impressed  e.m.f., 
the  current  in  receiver  circuit  is 

1=       ^° 
r  +  jxo 

the  current  in  condenser  circuit  is 


I  = 


E0 


JXc 

and  the  total  current  is 
70  =  /  +  7i  =  EQ 


I 


1 


i"  -r  JXQ      jxc 

2    ~J 


r2  +  XQ 


4)1 


or,  in  absolute  terms, 
/o 


-*Jt 


while  the  e.m.f.  at  receiver  terminals  is 

r 


E  =  IT  =  E< 


E  = 


E0r 


62.  The  main  current,  70,  is  in  phase  with  the  impressed 
e.m.f.,  EQ,  or  the  lagging  current  is  completely  balanced,  or 
supplied  by,  the  condensive  reactance,  if  the  imaginary  term  in 
the  expression  of  J0  disappears;  that  is,  if 


-       =  0. 

xc 


74  ALTERNATING-CURRENT  PHENOMENA 

This  gives,  expanded, 

_  r2  +  xo2 


Hence  the  capacity  required  to  compensate  for  the  lagging 
current  produced  by  the  insertion  of  inductive  reactance  in 
series  with  a  non-inductive  circuit  depends  upon  the  resistance 
and  the  inductive  reactance  of  the  circuit.  XQ  being  constant, 
with  increasing  resistance,  r,  the  condensive  reactance  has  to  be 
increased,  or  the  capacity  decreased,  to  keep  the  balance. 

Substituting 

r2  +  *o2 

Xc   =   -  —  > 

XQ 

we  get,  as  the  equations  of  the  inductive  circuit  balanced  by 
condensive  reactance, 

EQ  E0(r  -  jxQ) 


EQr 
= 


and  for  the  power  expended  in  the  receiver  circuit, 


r2  +  zo2 

that  is,  the  main  current  is  proportional  to  the  expenditure  of 
power. 

For  r  =  0,  we  have  xc  =  x0,  as  the  condition  of  balance. 

Complete  balance  of  the  lagging  component  of  current  by 
shunted  capacity  thus  requires  that  the  condensive  reactance  xc 
be  varied  with  the  resistance,  r;  that  is,  with  the  varying  load 
on  the  receiver  circuit. 

In  Fig.  66  are  shown,  for  a  constant  impressed  e.m.f.,  E0  = 
1000  volts,  and  a  constant  series  reactance,  XQ  =  100  ohms,  values 
for  the  balanced  circuit  of 

current  in  receiver  circuit      (Curve  I), 
current  in  condenser  circuit  (Curve  II), 
current  in  main  circuit  (Curve  III), 

e.m.f.  at  receiver  terminals   (Curve  IV), 


CIRCUITS  CONTAINING  RESISTANCE 


75 


with  the  values  the  resistance,  r,  of  the  receiver  circuit  as 
abscissas. 

63.  If,  however,  the  condensive  reactance  is  left  unchanged, 
xc  =  x0  at  the  no-load  value,  the  circuit  is  balanced  for  r  =  0, 
but  will  be  overbalanced  for  r>0,  and  the  main  current  will  be- 
come leading. 


IMPRESSED  E.Jyi.F.  CONSTANT,  E0=IOOO  VOCTS. 
SERIES  REACTANCE  CONSTANT,  Xo=IOO  OHMS. 
VARIABLE  RESISTANCE  IN  RECEIVER  CIRCUIT. 

BALANCED  BY  VARYIN.G  THE  SHUNTED  CONDEN3ANCE. 

I.   CURRENT  IN  RECEIVER  CIRCUIT. 
(I.  CURRENT  IN  CONDENSER  CIRCUIT. 

III.  CURRENT  IN  MAIN  CIRCUIT. 

IV.  E.M.F.  AT  RECEIVER  CIRCUIT. 


10  JiO  30  40  60  60  70  80  90  100  110  120  130  140  150  160  170  ISO  190  200 

FIG.  66. — Compensation  of  lagging  currents  in  receiving  circuit  by  variable 
shunted  condensance. 


We  get  in  this  case, 
E0 


xc  =  xQ; 


r  -t-  JXQ 
The  difference  of  phase  in  the  main  circuit  is 


tan  00  = ' 

XQ 


76 


ALTERNATING-CURRENT  PHENOMENA 


which  is  =  0,  when  r  =  0  or  at  no-load,  and  increases  with 
increasing  resistance,  as  the  lead  of  the  current.  At  the  same 
time,  the  current  in  the  receiver  circuit,  /,  is  approximately  con- 
stant for  small  values  of  r,  and  then  gradually  decreases. 

In  Fig.  67  are  shown  the  values  of  7,  /i,  70,  E,  in  Curves 
I,  II,  III,  IV,  similarly  as  in  Fig.  60,  for  EQ  =  1000  volts, 
xc  =  xQ  =  100  ohms,  and  r  as  abscissas. 


AMPERES 

IMPRESSED  E.M.F.  CONSTANT,  EO—  1000  VOLTS. 
SERIES  REACTANCE  CONSTANT,  £o=tOO  OH  MB. 
SHUNTED  CONDENSANCE  CONSTANT,  C=  IOO  OHMS, 
VARIABLE  RESISTANCE  IN  RECEIVED  CIRCUIT- 
'(.CURRENT  IN  RECEIVER  CIRCUIT. 

II.  CURRENT  IN  CONDENSER  CIRCUIT. 
III.  CURRENT  IN  MAIN  CIRCUIT. 
IV.E.M.F.  AT  RECEIVER  CIRCUIT' 

VOL 

s 

u. 

1 

8 
7 
6 

6 

1 
8- 

10CK 
900 

IN 

~-^, 

~~  —  , 

^ 

"^ 

**Xs 

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^-- 

—  • 

•• 

—  —  - 

^~- 

,-      ••• 

T= 

—   — 

wo 
coo 
soo 

-400 
-300 
.200 

"^ 

4^ 

<-• 

,  

. 

x- 

^^ 

^*^^ 

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— 

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s 

-~^. 

•  — 

"-  '  . 

% 

^ 

'  

""—-»* 

—  '     —. 

—•i.^. 

/ 

/ 

\ 

/ 

RESISTANCE  7*—  «-OF  RECEIVER  CIRCUIT.  OHMS. 

/ 

1    1    1    1    1    1    II    1    1    1 

10     SO    SO    40     60    60     70    80    90    100  110  J20  130  140  150  WO  170  U 

0  190  800  OHMS 

FIG.  67. 

5.  Constant  Potential — Constant-current  Transformation 

64.  In  a  constant  potential  circuit  containing  a  large  and 
constant  reactance,  XQ,  and  a  varying  resistance,  r,  the  current 
is  approximately  constant,  and  only  gradually  drops  off  with 
increasing  resistance,  r — that  is,  with  increasing  load — but 
the  current  lags  greatly  behind  the  voltage.  This  lagging  current 
in  the  receiver  circuit  can  be  supplied  by  a  shunted  condensance. 
Leaving,  however,  the  condensance  constant,  xc  =  XQ,  so  as  to 
balance  the  lagging  current  at  no-load,  that  is,  at  r  =  0,  it  will 
overbalance  with  increasing  load,  that  is,  with  increasing  r,  and 
thus  the  main  current  will  become  leading,  while  the  receiver 
current  decreases  if  the  impressed  voltage,  E0,  is  kept  constant. 
Hence,  to  keep  the  current  in  the  receiver  circuit  entirely  con- 
stant, the  impressed  voltage,  E0,  has  to  be  increased  with  in- 


CIRCUITS  CONTAINING  RESISTANCE  77 

creasing  resistance,  r;  that  is,  with  increasing  lead  of  the  main 
current.  Since,  as  explained  before,  in  a  circuit  with  leading 
current,  a  series  inductive  reactance  raises  the  potential,  to 
maintain  the  current  in  the  receiver  circuit  constant  under  all 
loads,  an  inductive  reactance,  x2,  inserted  in  the  main  circuit, 
as  shown  in  the  diagram,  Fig.  68,  can  be  used  for  raising  the 
voltage,  EQ,  with  increasing  load,  and  by  properly  choosing  the 
inductive  and  the  condensive  reactances,  practically  constant 
current  at  varying  load  can  be  produced  from  constant  voltage 
supply,  and  inversely. 


FIG.  68 

The  generation  of  alternating-current  electric  power  almost 
always  takes  place  at  constant  potential.  For  some  purposes, 
however,  as  for  operating  series  arc  circuits,  and  to  a  limited 
extent  also  for  electric  furnaces,  a  constant,  or  approximately 
constant,  alternating  current  is  required. 

Such  constant  alternating  currents  can  be  produced  from 
constant  potential  circuits  by  means  of  inductive  reactances, 
or  combinations  of  inductive  and  condensive  reactances;  and 
the  investigation  of  different  methods  of  producing  constant 
alternating  current  from  constant  alternating  potential,  or 
inversely,  constitutes  a  good  illustration  of  the  application  of 
the  terms  "  impedance,"  "  reactance,"  etc.,  and  offers  a  large 
number  of  problems  or  examples  for  the  application  of  the 
method  of  complex  quantities.  A  number  of  such  are  given  in 
"Theory  and  Calculation  of  Electric  Circuits." 


CHAPTER  X 

RESISTANCE  AND  REACTANCE  OF  TRANSMISSION 

LINES 

65.  In  alternating-current  circuits,  voltage  is  consumed  in 
the  feeders  of  distributing  networks,  and  in  the  lines  of  long- 
distance transmissions,  not  only  by  the  resistance,  but  also 
by  the  reactance,  of  the  line.  The  voltage  consumed  by  the 
resistance  is  in  phase,  while  the  voltage  consumed  by  the  react- 
ance is  in  quadrature,  with  the  current.  Hence  their  in- 
fluence upon  the  voltage  at  the  receiver  circuit  depends  upon 
the  difference  of  phase  between  the  current  and  the  voltage  in 
that  circuit.  As  discussed  before,  the  drop  of  potential  due  to 
the  resistance  is  a  maximum  when  the  receiver  current  is  in 
phase,  a  minimum  when  it  is  in  quadrature,  with  the  voltage. 
The  change  of  voltage  due  to  line  reactance  is  small  if  the 
current  is  in  phase  with  the  voltage,  while  a  drop  of  potential  is 
produced  with  a  lagging,  and  a  rise  of  potential  with  a  leading, 
current  in  the  receiver  circuit. 

Thus  the  change  of  voltage  due  to  a  line  of  given  resistance 
and  reactance  depends  upon  the  phase  difference  in  the  receiver 
circuit,  and  can  be  varied  and  controlled  by  varying  this  phase 
difference;  that  is,  by  varying  the  admittance,  Y  =  g  —  jb,  of 
the  receiver  circuit. 

The  conductance,  g,  of  the  receiver  circuit  depends  upon 
the  consumption  of  power — that  is,  upon  the  load  on  the 
circuit — and  thus  cannot  be  varied  for  the  purpose  of  regu- 
lation. Its  susceptance,  6,  however,  can  be  changed  by  shunt- 
ing the  circuit  with  a  reactance,  and  will  be  increased  by  a 
shunted  inductive  reactance,  and  decreased  by  a  shunted  con- 
densive  reactance.  Hence,  for  the  purpose  of  investigation,  the 
receiver  circuit  can  be  assumed  to  consist  of  two  branches,  a 
conductance,  g, — the  non-inductive  part  of  the  circuit — 
shunted  by  a  susceptance,  6,  which  can  be  varied  without 
expenditure  of  energy.  The  two  components  of  current  can 
thus  be  considered  separately,  the  energy  component  as  deter- 

78 


TRANSMISSION  LINES  79 

mined  by  the  load  on  the  circuit,  and  the  wattless  component, 
which  can  be  varied  for  the  purpose  of  regulation. 

Obviously,  in  the  same  way,  the  voltage  at  the  receiver  circuit 
may  be  considered  as  consisting  of  two  components,  the  power 
component,  in  phase  with  the  current,  and  the  wattless  com- 
ponent, in  quadrature  with  the  current.  This  will  correspond 
to  the  case  of  a  reactance  connected  in  series  to  the  non-inductive 
part  of  the  circuit.  Since  the  effect  of  either  resolution  into 
components  is  the  same  so  far  as  the  line  is  concerned,  we  need 
not  make  any  assumption  as  to  whether  the  wattless  part  of  the 
receiver  circuit  is  in  shunt,  or  in  series,  to  the  power  part. 

Let 

ZQ  =  r0  -f-  J£o    =  impedance  of  the  line; 

Y  =  g  —  jb      =  admittance  of  receiver  circuit; 

y  =  vV  +  &2; 

EQ  =  e0  -\-  je'o  =  impressed  voltage  at  generator  end  of  line ; 


+  e0'2; 
E  =  e  +  jef     =  voltage  at  receiver  end  of  line; 

E  =  Ve2  +  e'2;      , 
/o  =  *o  -f-  ji\    =  current  in  the  line  ; 

/o  =  Vio2  +  *o'2. 
The  simplest  condition  is  the  non-inductive  circuit. 

1.  Non-inductive  Receiver  Circuit  Supplied  over  an  Inductive 

Line 

66.  In  this   case,   the  admittance  of  the  receiver  circuit  is 
F  =  g,  since  6  =  0. 

We  have  then 

current,  IQ  =  Eg; 

impressed  voltage:        E0  =  E  +  Z0I0  =  E(l  -f  Z0g). 

Hence  —  voltage  at  receiver  circuit, 


1  +  Z<*g 
current, 

°  ~ 


ZQg 


80  ALTERNATING-CURRENT  PHENOMENA 

Hence,  in  absolute  values  —  voltage  at  receiver  circuit, 

is- 

current, 

" 


The  ratio  of  e.m.fs.  at  receiver  circuit  and  at  generator,  or 
supply  circuit,  is 

E  1 


and  the  power  delivered  in  the  non-inductive  receiver  circuit,  or 
output, 


7  F  - 
= 


(1  +  grtf  +  fxf 

As  a  function  of  g,  and  with  a  given  J00,  r0,  and  XQ)  this  power 
is  a  maximum,  if 

dP 

¥  =  0; 

that  is, 

-  1  +  grVo2  +  gV  =  0; 
hence, 

conductance  of  receiver  circuit  for  maximum  output, 

1  1 


Resistance  of  receiver  circuit,  rm  =  —  —  ZQ; 


and,  substituting  this  in  P, 
Maximum  output,     Pm 


-^—f  —  -.  —  r  =  —  r  -  7  —  - 

2  (r0  +  to)      2  {TO  +  Vro2  +  ^o2 

and  ratio  of  e.m.f.  at  receiver  and  at  generator  end  of  line, 


efficiency, 

rm      r0       r0 

That  is: 

The  output  which  can  be  transmitted  over  an  inductive  line  of 
resistance,  r0,  and  reactance,  x0 — that  is,  of  impedance,  z0 — into  a 


TRANSMISSION  LINES 


81 


non-inductive  receiver  circuit,  is  a  maximum  if  the  resistance  of  the 
receiver  circuit  equals  the  impedance  of  the  line,  r  =  ZQ, 
and  is 


____ 
2  (r0  +  zo) 

The  output  is  transmitted  at  the  efficiency  of 


and  with  a  ratio  of  e.m.fs.  of 


(•+3 


NON-INDUCTIVE  RECE 
SUPPLIED  OVER  INQUCTIV 

AND  OVER  NON-INDUCTIVE 

n  =i 

CURVE  1.    E.  M.  F.  AT  RECEIVER 

M         IV.               M             "                  )' 

"       II-   OUTPUT  IN            >' 

JVER  C  RCU  T 
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CIRCUIT,  INDUCTIVE  U 
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0   10  20  30  40  50  60  70  80  tK)  100  110  120  130  140  150  160  170  180 

FIG.  69. — Non-inductive  receiver-circuit  supplied  over  inductive  line. 

67.  We  see  from  this  that  the  maximum  output  which  can 
be  delivered  over  an  inductive  line  is  less  than  the  output  de- 
livered over  a  non-inductive  line  of  the  same  resistance — 
that  is,  which  can  be  delivered  by  continuous  currents  with  the 
same  generator  potential.  / 

In  Fig.  69  are  shown,  for  the  constants, 


82  ALTERNATING-CURRENT  PHENOMENA 

Eo  =  1000  volts,  Z0  =  2.5  -pGj;  that  is,  r0  =  2.5  ohms,  X0  =  6 
ohms,  20  =  6.5  ohms,  with  the  current  70  as  abscissas,  the  values. 

e.m.f  .  at  receiver  circuit,  E,  (Curve  I)  ; 

output  of  transmission,  P,  (Curve  II)  ; 

efficiency  of  transmission,  (Curve  III). 

The  same  quantities  for  a  non-inductive  line  of  resistance, 
r0  =  2.5  ohms,  XQ  =  0,  are  shown  in  Curves  IV,  V,  and  VI. 

2.  Maximum  Power  Supplied  over  an  Inductive  Line 

68.  If  the  receiver  circuit  contains  the  susceptance,  b,  in 
addition  to  the  conductance,  g,  its  admittance  can  be  written 
thus: 

Y  =  g  -  jb,  y  =  vV  +  b2. 

Then,  current,  /o  =EY\ 

Impressed  voltage,  E0  =  E  +  I0Z0  =  E(l  +  7Z0). 

Hence,  voltage  at  receiver  terminals, 


_ 

'  1  +  YZ0       (1  +  rog  +  x<*b)  +  j(x«g  -  r0b)  ' 
current, 

EQY     =  _  E,(g-jV)  _  . 
'  1  +  KZo       (1  +  ro0  +  xQb)  +j(x0g  -  rob)' 
or,  in  absolute  values,  voltage  at  receiver  circuit, 


current, 

JO    —    ^  0  A  /  ~7^      , , T\~S     I      7  T\z ' 

ratio  of  e.m.fs.  at    receiver  circuit    and   at    generator  circuit, 


W  +  xJ>Y  +  (x0g  - 
and  the  output  in  the  receiver  circuit  is 

P  =  E*g  =  EQWg. 

69.  (a)   Dependence  of  the  output  upon  the  susceptance  of  the 
receiver  circuit. 

At  a  given  conductance,  g,  of  the  receiver  circuit,  its  output, 
P  =  E02a2g,  is  a  maximum  if  a2  is  a  maximum;  that  is,  when 


s  a  mnmum. 


TRANSMISSION  LINES  83 

The  condition  necessary  is 

i=°- 

or,  expanding, 

zo(l  +  r0g  +  x0b)  -  r0(xQg  -  r06)  =  0. 
Hence 
Susceptance  of  receiver  circuit, 

Xo2  X0 

-=     ~~  =      "&° 


or  6  +  60  =  0, 

that  is,  if  the  sum  of  the  susceptances  of  line  and  of  receiver 

circuit  equals  zero. 

Substituting  this  value,  we  get 
ratio  of  e.m.fs.  at  maximum  output, 

E_  1 

ai       E0      *<*  +  *)' 

maximum  output, 


current, 

, 
" 


-.*$ 

and,  since, 


T  + 


?V 
it  is, 


r0 


2        2 


=  Z02  to  +  0o)2, 
Thus,  it  is,  current, 

~fco2 


84  ALTERNATING-CURRENT  PHENOMENA 

phase  difference  in  receiver  circuit, 

b           60 
tan  B  =  -  = ; 

g  g 

phase  difference  in  generator  circuit, 

tan  0o  =  ~        ~  —  s~~i 5" 

r  +  r0        </o2r  +  <72/o 

70.  (6)  Dependence  oj  the  output  upon  the  conductance  of  the 
receiver  cirwit. 

At  a  given  susceptance,  6,  of  the  receiver  circuit,  its  output, 
P  =  E02  a2g,  is  a  maximum  if 


or 

(I  +  ro0  +  ^o?>)2  +  (xQg  -  r0b) 


2\  _ 

/  = 


that  is,  expanding, 

(1  +  r0g  +  xob)2  +  (xofir  -  r0fc)2  -  2  g(r0  +  r02g  +  x0W  =  0; 
or,  expanding, 

(b  +  60)2  =  g2  -  sfo2;    flf  =  Vgo*  +  (b  +  60)  2. 

Substituting  this  value  in  the  equation  for  a,  §68,  we  get  — 
ratio  of  e.m.fs., 

1 


£o 


Y2J002  +  (6  +  60) s  +  SroV^o2  +  (b  +  60)2} 


z0  V2  g(g  +  00)       V2  0(0  +  g0) ' 
power, 

^o22/o2  E02y02 


\ 

As  a  function  of  the  susceptance,  b,  this  power  becomes  a 
maximum  for  -3=-  =  0,  that  is,  according  to  §69  if 

6  =  -  60. 
Substituting  this  value,  we  get 

b  =  —  60,  g  =  go,  y  =  2/o,  hence:  Y  =  g  —  jb  =  g0  +  j&oj 
x  =  —  x0,  r  =  r0,  z  =  20>  Z  =  r  +  jx  =  r0  —  jx0; 


TRANSMISSION  LINES  85 

substituting  this  value,  we  get — 

ratio  of  e.m.fs..  am  =  pr^-  =  ^-; 

2  go       2rV 


power 


that  is,  the  same  as  with  a  continuous-current  circuit;  or,  in 
other  words,  the  inductive  reactance  of  the  line  and  of  the 
receiver  circuit  can  be  perfectly  balanced  in  its  effect  upon  the 
output. 

71.  As  a  summary,  we  thus  have: 

The  output  delivered  over  an  inductive  line  of  impedance, 
ZQ  =  rQ  +  jx0,  into  a  non-inductive  receiver  circuit,  is  a  maxi- 
mum for  the  resistance,  r  =  z0,  or  conductance,  g  =  2/0,  of  the 
receiver  circuit,  and  this  maximum  is 


2  (r0  +  z0) 
at  the  ratio  of  voltages, 


With  a  receiver  circuit  of  constant  susceptance,  6,  the  out- 
put, as  a  function  of  the. conductance,  g,  is  a  maximum  for  the 
conductance, 

and  is 

P  _       #o22/o2 

=  2(</  +  00)' 
at  the  ratio  of  voltages, 


With  a  receiver  circuit  of  constant  conductance,  g,  the  output,  as 
a  function  of  the  susceptance,  6,  is  a  maximum  for  the  susceptance 
6  =  —  60,  and  is 


•*•  2  /          I  \2 

ZQ  (g  T  0o) 

at  the  ratio  of  voltages, 

=          1 

~  z0(g  +  go) 


86 


ALTERNATING-CURRENT  PHENOMENA 


The  maximum  output  which  can  be  delivered  over  an  induc- 
tive line,  as  a  function  of  the  admittance  or  impedance  of  the 
receiver  circuit,  takes  place  when  Z  =  r0  —  jx0,  or  Y  =  go  -f  j60; 
that  is,  when  the  resistance  or  conductance  of  receiver  circuit 
and  line  are  equal,  the  reactance  or  susceptance  of  the  receiver 

7?  2 

circuit  and  line  are  equal  but  of  opposite  sign,  and  is  P  =  T— > 

4  r  o 

or  independent  of  the  reactances,  but  equal  to  the  output  of  a 


.01   .02   .03    .04    .05    .06  ,07     .08    .09  .10    .11    .12    .13 


FIG.  70.  —  Variation  of  the  potential  in  line  at  different  loads. 

continuous-current  circuit  of  equal  line  resistance.     The  ratio 
of  voltages  is,   in  this  case,   a  =  ~  —  ,   while  in  a  continuous- 

4  TQ 

current  circuit  it  is  equal  to  0.5.     The  efficiency  is  equal  to  50 
per  cent. 

72.  As  an  example,  in  Fig.  70  are  shown  for  the  constants 

EQ  =  1000  volts,  and        Z0  =  2.5  +  6j;  that  is,  for 
r0  =  2.5  ohms,  XQ  =  6  ohms,          z0  =  6.5  ohms, 


TRANSMISSION  LINES 


87 


and  with  the  variable  conductances  as  abscissas,   the  values 
of  the 

output,     .  in  Curve  I,  Curve  III,  and  Curve  V; 

ratio  of  voltages,  in  Curve  II,  Curve  IV,  and  Curve  VI; 

Curves  I  and  II  refer  to  a  non-inductive  receiver 

circuit ; 
Curves  III  and  IV  refer  to  a  receiver  circuit  of 

constant  susceptance b  =  0.142 


OUTPUT  P  AND  RATIO  OF  POTENTIAL  d  AT  RECEI  VINO-AN  D 
.SENDING  END  OF  LINE  OF  IMPEDANCR  Z0=3.5+  «'  _ 

AT  CONSTANT  IMPRESSED  E.M.F. 


I   OUTPUT 
II   RATIO  OF  POTENTIALS 


\\ 


\ 


SUSCERTANCE  OF  RECEIVER 


CIRCl  IT 


-.3       -.2       -.1  +.1       -f.2       +.3       +.4 

FIG.  71. — Variation  of  the  potential  in  line  at  various    loads. 


Curves  V  and  VI  refer  to  a  receiver  circuit  of 

constant  susceptance b  =  —  0.142 

Curves  VII  and  VIII  refer  to  a  non-inductive  re- 
ceiver circuit  and  non-inductive  line. 
In  Fig.  71  the  output  is  shown  as  Curve  I,   and   the  ratio 

of  voltages   as  Curve  II,  for  the  same  line    constants,  for  a 

constant  conductance,  g  =  0.0592  ohm,  and  for  variable  sus- 

ceptances,  6,  of  the  receiver  circuit. 


88  ALTERNATING-CURRENT  PHENOMENA 

3.  Maximum  Efficiency 

73.  The  output  for  a  given  conductance,  g,  of  a  receiver 
circuit  is  a  maximum  if  6  =  —  60.  This,  however,  is-  generally 
not  the  condition  of  maximum  efficiency. 

The  loss  of  power  in  the  line  is  constant  if  the  current  is 
constant;  the  output  of  the  generator  for  a  given  current  and 
given  generator  voltage  is  a  maximum  if  the  current  is  in  phase 
with  the  voltage  at  the  generator  terminals.  Hence  the  con- 
dition of  maximum  output  at  given  loss,  or  of  maximum  effi- 
ciency is 

tan  60  =  0. 
The  current  is 

77?  77T 

,      _  &0  _     -PO . 

.  °  ~  Z  +  Z0  ~  (r  +  r0)  +  j(x  +  x0)  ' 

The  current,  /o,  is  in  phase  with  the  e.m.f.,  EQ,  if  its  quad- 
rature component — that  is,  the  imaginary  term — disappears, 
or 

x  +  x0  =  0. 

This,  therefore,  is  the  condition  of  maximum  efficiency, 

x  =  —  xQ. 

Hence,  the  condition  of  maximum  efficiency  is  that  the 
reactance  of  the  receiver  circuit  shall  be  equal,  but  of  opposite 
sign,  to  the  reactance  of  the  line. 

Substituting  x  =  —  XQ,  we  have: 
ratio  of  e.m.fs., 

E_  z  Vr2  +  a02, 

power, 


and  depending  upon  the  resistance  only,   and  not  upon   the 
reactance. 

This  power  is  a  maximum  if  g  =  go,  as  shown  before;  hence, 
substituting  g  =  gQ,  r  —  r0, 

p1  2 

maximum  power  at  maximum  efficiency,  Pm  =  T^~> 

ftrso 

at  a  ratio  of  potentials,  am  =  « — ' 
or  the  same  result  as  in  §70. 


TRANSMISSION  LINES 


89 


In  Fig.  72  are  shown,  for  the  constants, 
E0  =  100  volts, 

Z0  =  2.5  +  6j;  r0  =  2.5  ohms,  XQ  =  6  ohms,  z0  =  6.5  ohms, 
and  with  the  variable  conductances,  g,  of  the  receiver  circuit 
as  abscissas,  the 

Output  at  maximum  efficiency,  (Curve  I)  ; 

Volts  at  receiving  end  of  line,      (Curve  II); 

Efficiency  =  — £ — ,  (Curve  III). 


FIG. 


.01  .02  .03          .04  .05          .00          .O/  .08 

72.  —  Load  characteristics  of  transmission  lines. 


4.  Control  of  Receiver  Voltage  by  Shunted  Susceptance 

74.  By  varying  the  susceptance  of  the  receiver  circuit,  the 
voltage  at  the  receiver  terminals  is  varied  greatly.  Therefore, 
since  the  susceptance  of  the  receiver  circuit  can  be  varied  at 
will,  it  is  possible,  at  a  constant  generator  voltage,  to  adjust 
the  receiver  susceptance  so  as  to  keep  the  voltage  constant  at 
the  receiver  end  of  the  line,  or  to  vary  it  in  any  desired  manner, 
and  independently  of  the  generator  voltage,  within  certain  limits. 


90          ALTERNATING-CURRENT  PHENOMENA 

The  ratio  of  voltages  is 

E  1 


a  =  -JJT  —  — /  • 

^o       V  (1  -f  r0g  +  x0b  )2  +  (xQg  —  r06)2 

If  at  constant  generator  voltage  E0  the  receiver  voltage  E 
shall  be  constant, 

a  =  constant; 
hence, 

(1  -f  rQg  +  z 
or,  expanding, 

6  =  -  bo  - 

which  is  the  value  of  the  susceptance,  6,  as  a  function  of  the 
receiver  conductance — that  is,  of  the  load — which  is  required 
to  yield  constant  voltage,  aEQ,  at  the  receiver  circuit. 

For  increasing  g,  that  is,  for  increasing  load,  a  point  is  reached 
where,  in  the  expression 

b  =  —  b0  4-  \  (—}    —  (a  +  a0)2, 


the  term  under  the  root  becomes  negative,  and  b  thus  imaginary, 
and  it  thus  becomes  impossible  to  maintain  a  constant  voltage, 
aEo.  Therefore  the  maximum  output  which  can  be  transmitted 
at  voltage,  aE0,  is  given  by  the  expression 


-  (9  +  ff»)2  =  0; 
hence  the  susceptance  of  receiver  circuit  is  b  =  —  b0,  and   the 

conductance  of  receiver  circuit  is  g  =  —  gQ  +      > 

a 


-  00  ,  the  output. 

75.  If  a  =  1,  that  is,  if  the  voltage  at  the  receiver  circuit 
equals  the  generator  voltage, 

g  =  2/o  -  go',  P  =  Eo^yo  -  g0). 
If      •  a  =  1,    when  g  =  0,       b  =  0 

when  g  >  0,       b  <  0; 
if  a  >  1,    when  g  =  0,  or  g  >  0,  b  <  0, 

that  is,  condensive  reactance; 
if  a  <  1,    when  g  =  0,         b  >  0, 


TRANSMISSION  LINES  91 


whensr=  -sro  +  A-^2,     6  =  0; 


when  g  >  -  gQ  +  -  602,     6  <  0, 

or,  in  other  words,  if  a  <  1,  the  phase  difference  in  the  main 
line  must  change  from  lag  to  lead  with  increasing  load. 

76.  The  value  of  a  giving  the  maximum  possible  output  in 

a  receiver  circuit  is  determined  by  -7—  =  0; 


expanding  2  a  (^  -  <7o)  -       T  =  0; 

hence  yo  =  2  ago, 

2/o  1  Zo 


= 


the  maximum  output  is  determined  by 

7/0 

g  =  -  0o  -I-  —  =  gQ'} 

2 
and  is,  P 

From  a  = 


20o       2r0 

the  line  reactance,  XQ,  can  be  found,  which  delivers  a  maximum 
output  into  the  receiver  circuit  at  the  ratio  of.  voltages,  a,  as 

ZQ  =  2  rQa, 
XQ  =  r0\/4  a2  —  1  ; 
for  a  =  1, 

Zo  =  2r0; 


If,  therefore,  the  line  impedance  equals  2  a  times  the  line 

E  2 

resistance,  the  maximum  output,  P  =  j—  ,  is  transmitted  into 

tb  7*o 

the  receiver  circuit  at  the  ratio  of  voltages,  a. 

If  ^0  =  2r0,  or  XQ  =  r0\/3>  the  maximum  output,  P  = 


fi1  2 


can  be  supplied  to  the  receiver  circuit,  without  change  of  voltage 
at  the  receiver  terminals. 

Obviously,  in  an  analogous  manner,  the  law  of  variation 
of  the  susceptance  of  the  receiver  circuit  can  be  found  which 
is  required  to  increase  the  receiver  voltage  proportionally  to 


92 


ALTERNATING-CURRENT  PHENOMENA 


the  load;  or,  still  more  generally,  to  cause  any  desired  varia- 
tion of  the  voltage  at  the  receiver  circuit  independently  of  any 
variation  of  the  generator  voltage,  as,  for  instance,  to  keep  the 
voltage  of  a  receiver  circuit  constant,  even  if  the  generator  volt- 
age fluctuates  widely. 

77.  In  Figs.   73,   74,   and   75   are  shown,   with   the  output, 
P  =  E0*ga*,   as   abscissas,    and   a  constant  impressed  voltage, 


RATIO  OF  RECEIVER  VOLTAGE  TO  SENDER  VOLTAGE:  O  =  1.0 
LINE.IMPEDANCE*  Zo=2-« +6j 
I  ENERGY  CURRENT  CONSTANT  GENERATOR  POTENTIAL  E  0= 

II  REACTIVE  CURRENT 

III  TOTAL  CURRENT 

IV  CURRENT  IN  NON-INDUCTIVE  RECEIVER  CIRCUIT  WITHOUT  COMPENSATION 

V  POTENTIAL      n  ,_,  n  n  «  ,_, 


100 


20          80  40  50          60  70  80  90 

OUTPUT  IN  RECEIVER  CIRCUIT,  KILOWATTS 

FIG.  73.  —  Variation  of  voltage  of  transmission  lines. 

EQ  =  1000  volts,  and  a  constant  line  impedance,  ZQ  =  2.5  +  6  j, 
or  r0  =  2.5  ohms,  x0  =  6  ohms,  z  =  6.5  ohms,  the  following 
values  : 

power  component  of  current,  gE,  (Curve  I)  ; 

reactive,  or  wattless  component  of  current,  bE,  (Curve  II)  ; 
total  current,  yE,  (Curve  III), 

and  power  factor  at  generator  for  the  following  conditions: 

a  =  1.0  (Fig.  73);  a  =  0.7  (Fig.  74);   a  =  1.3  (Fig.  75). 

For  the  non-inductive  receiver  circuit  (in  dotted  lines),  the 
curve  of  e.m.f.,  E,  and  of  the  current,  I  =  gE,  are  added  in  the 
three  diagrams  for  comparison,  as  Curves  IV  and  V. 

As  shown,  the  output  can  be  increased  greatly,  and  the 
voltage  at  the  same  time  maintained  constant,  by  the  judicious 


TRANSMISSION  LINES 


93 


RATIO  OF  RECEIVER  VOLTAGE  TO  BENDER  VOLTAGE:  a  =  .7 
LINE  IMPEDANCE:  Z 5=2. B  + -6 j 
I     ENERGY  CURRENT  CONSTANT  GENERATOR  POTENTIAL  E, 

II  REACTIVE  CURRENT 

III  TOTAL  CURRENT 

IV  POTENTIAL  IN  NON-INDUCTIVE  CIRCUIT  WITHOUT  COMPENSATION 


80          40          60  60  70  8( 

OUTPUT  IN  RECEIVER  CIRCUIT,  KILOWATTS 


240 

220 
200 

180 
100 
140 
120 
100 
80 
60 
40 
20 
0 
20 
40 


FIG.  74. — Variation  of  voltage  of  transmission  lines. 


RATIO  OF  RECEIVER  VOLTAGE  TO  SENDER  VOLTAGE:  a  =  1,8 
LINE  IMPEDANCE:  Z0=2.6  -f-  OJ 

I  ENERGY  CURRENT  CONSTANT  GENERATOR  POTENTIAL 

II  REACTIVE  CURRENT 

III  TOTAL  CURRENT 

IV  POTENTIAL  IN  NON-INDUCTIVE  RECEIVER  CIRCUIT  WITHOUT  COMPENSATION 


Eo=r100( 


OUTPUT  IN  RECEIVER  CIRCUIT,  KILOWATTS 

FIG.  75. — Variation  of  voltage  of  transmission  lines. 


240 

220 


180 


100 


94  ALTERNATING-CURRENT  PHENOMENA 

use  of  shunted  reactance,  so  that  a  much  larger  output  can  be 
transmitted  over  the  line  with  no  drop,  or  even  with  a  rise,  of 
voltage.  Shunted  susceptance,  therefore,  is  extensively  used 
for  voltage  control  of  transmission  lines,  by  means  of  synchronous 
condensers,  or  by  synchronous  converters  with  compound  field 
winding. 

5.  Maximum  Rise   of  Voltage  at  Receiver  Circuit 

78.  Since,  under  certain  circumstances,  the  yoltage  at  the 
receiver  circuit  may  be  higher  than  at  the  generator,  it  is  of 
interest  to  determine  what  is  the  maximum  value  of  voltage,  E, 
that  can  be  produced  at  the  receiver  circuit  with  a  given  generator 
voltage,  EQ. 

The  condition  is  that 

1 


maximum  or  —  ^  =  minimum: 
a2 


that  is, 


dg  db 

substituting, 

and  expanding,  we  get, 


___     f\  m  

dg  202 

— a  value  which  is  impossible,  since  neither  r0  nor  g  can  be 
negative.     The  next  possible  value  is  g  =  0 — a  wattless  circuit. 
Substituting  this  value,  we  get, 


and  by  substituting,  in 

—4—  =  0,     b  =  -  ^\  =  -  60, 
do  2o 

b  +  60  =  0; 

that  is,  the  sum  of  the  susceptances  =  0,  or  the  inductive  sus- 
ceptance of  the  line  is  balanced  by  the  capacity  susceptance  of 
the  load. 


TRANSMISSION  LINES 


95 


.  Substituting 
we  have 


b  =  -  60, 
1  zo 


a.  = 


r° 


The  current  in  this  case  is 
I  = 


VOLT 

\ 

s 

\ 

I 

e 

s 

\ 

\ 

1900 
1800 
1700 
1600 
1500 
1400 
1300 
1200 
1100 
1000 
800 
800 
700 
000 
500 
400 
300 
200 
100 

o 

\, 

\ 

\ 

\ 

\N 

\ 

\\ 

CONSTANT  IMPRESSED  E.M.F.   E0=IOOO 
"          LINE  IMPEDANCE  Z0=2.54-6/ 
1    MAXIMUM  OUTPUT  BY  COMPENSATION 
II   MAXIMUM  EFFICIENCY  BY  COMPENSATION 
III   NON-JNDUCTIVE  RECEIVER  CIRCUIT 
IV   NON-INDUCTIVE  LINE  AND  NON-INDUCTIVE 
RECEIVER  CIRCUIT 

\\ 

\ 

I 

I 

a 

A 

*^ 

^x 

^* 

^ 

/u 

X 

•^ 

M 

^ 

X 

""^ 

\ 

{/ 

q1 

/ 

^ 

\ 

1 

/ 

J 

,^l 

// 

] 

X 

^ 

/ 

\ 

^ 

^ 

u 

k^S 

^ 

^ 

,  • 

^ 

^ 

*> 

OUT 

PUT 

K.V\ 

> 

1 

FIG.  76. — Efficiency  and  output  of  transmission  lines. 

or  somewhat  less  than  the  current  at  complete  resonance,  that 
is,  when  the  line  inductive  reactance,  XQ,  is  balanced  by  the 
capacity  reactance,  #,  of  the  load,  x  =  —  x0;  in  which  latter 
case  the  current  is 


96  ALTERNATING-CURRENT  PHENOMENA 

assuming  wattless  receiver  circuit,  and  is  in  phase  with  the 
voltage,  EQ. 

79.  As  summary  to  this  chapter,  in  Fig.  76  are  plotted,  for  a 
constant  generator  e.m.f.,  EQ  =  1000  volts,  and  a  line  impedance, 
Z0  =  2.5  +  6  j,  or  r0  =  2.5  ohms,  x0  =  6  ohms,  z0  =  6.5  ohms, 
and  with  the  receiver  output  as  abscissas  and  the  receiver 
voltages  as  ordinates,  curves  representing 

the  condition  of  maximum  output,  (Curve  I) ; 

the  condition  of  maximum  efficiency,  (Curve  II); 

the  condition  b  =  0,  or  a  non-inductive  receiver 

circuit,  (Curve  III); 

the   condition   6  =  0,    60  =  0,    or   a   non-inductive   line   and 
non-inductive  receiver  circuit. 

In  conclusion,  it  may  be  remarked  here  that  of  the  sources 
of  susceptance,  or  reactance, 

a  choking  coil  or  reactive  coil  corresponds  to  an  inductive 
reactance; 

a  condenser  corresponds  to  a  condensive  reactance; 

a  polarization  cell  corresponds  to  a  condensive  reactance; 

a  synchronous  machine  (motor,  generator  or  converter)  cor- 
responds to  an  inductive  or  a  condensive  reactance  at  will; 

an  induction  motor  or  generator  corresponds  to  an  inductive 
reactance. 

The  reactive  coil  and  the  polarization  cell  are  specially  suited 
for  series  reactance,  and  the  condenser  and  synchronous  machine 
for  shunted  susceptance. 


CHAPTER  XI 
PHASE  CONTROL 

80.  At  constant  voltage,  e0,  impressed  upon  a  circuit,  as  a 
transmission  line,  resistance,  r,  inserted  in  series  with  the  receiv- 
ing circuit,  causes  the  voltage,  ef  at  the  receiver  circuit  to  decrease 
with  increasing  current,  7,  through  the  resistance.  The  decrease 
of  the  voltage,  e,  is  greatest  if  the  current,  7,  is  in  phase  with 
the  voltage,  e — less  if  the  current  is  not  in  phase.  Inductive 
reactance  in  series  with  the  receiving  circuit,  e,  at  constant 
impressed  e.m.f.,  e0,  causes  the  voltage,  e,  to  drop  less  with  a 
unity  power-factor  current,  7,  but  far  more  with  a  lagging 
current,  and  causes  the  voltage,  e,  to  rise  with  a  leading 
current. 

While  series  resistance  always  causes  a  drop  of  voltage, 
series  inductive  reactance,  x,  may  cause  a  drop  of  voltage  or  a 
rise  of  voltage,  depending  on  whether  the  current  is  lagging  or 
leading.  If  the  supply  line  contains  resistance,  r,  as  well  as 
reactance,  x,  and  the  phase  of  the  current,  7,  can  be  varied 
at  will,  by  producing  in  the  receiver  circuit  lagging  or  leading 
currents,  the  change  of  voltage,  e,  with  a  change  of  load  in 
the  circuit  can  be  controlled.  For  instance,  by  changing  the 
current  from  lagging  at  no-load  to  lead  at  heavy  load  the 
reactance,  x,  can  be  made  to  lower  the  voltage  at  light  load 
and  raise  it  at  overload,  and  so  make  up  for  the  increasing  drop 
of  voltage  with  increasing  load,  caused  by  the  resistance,  r, 
that  is,  to  maintain  constant  voltage,  or  even  a  voltage,  e, 
which  rises  with  the  load  on  the  receiving  circuit,  at  constant 
voltage,  e0,  Sit  the  generator  side  of  the  line.  Or  the  wattless 
component  of  the  current  can  be  varied  so  as  to  maintain  unity 
power-factor  at  the  generator  end  of  the  line,  eQ,  etc. 

This  method  of  controlling  a  circuit  supplied  over  an  induc- 
tive line,  by  varying  the  phase  relation  of  the  current  in  the 
circuit,  has  been  called  "phase  control,"  and  is  used  to  a  great 
extent,  especially  in  the  transmission  of  three-phase  power  for 
conversion  to  direct  current  by  synchronous  converters  for 
7  97 


98  ALTERNATING-CURRENT  PHENOMENA 

railroading,  and  in  the  voltage  control  at  the  receiving  end  of 
very  long  high  voltage  transmission  lines. 

It  requires  a  receiving  circuit  in  which,  independent  of  the 
load,  a  lagging  or  leading  component  of  current  can  be  produced 
at  will.  Such  is  the  case  in  synchronous  motors  or  converters: 
in  a  synchronous  motor  a  lagging  current  can  be  produced  by 
decreasing,  a  leading  current  by  increasing,  the  field  excitation. 

81.  .If  in  a  direct-current  motor,  at  constant  impressed 
voltage,  the  field  excitation  and  therefore  the  field  magnetism  is 
decreased,  the  motor  speed  increases,  as  the  armature  has  to 
revolve  faster  to  consume  the  impressed  e.m.f.,  and  if  the  field 
excitation  is  increased,  the  motor  slows  down.  A  synchronous 
motor,  however,  cannot  vary  in  speed,  since  it  must  keep  in 
step  with  the  impressed  frequency,  and  if,  therefore,  at  constant 
impressed  voltage  the  field  excitation  is  decreased  below  that 
which  gives  a  field  magnetism,  that  at  the  synchronous  speed 
consumes  the  impressed  voltage,  the  field  magnetism  still  must 
remain  the  same,  and  the  armature  current  thus  changes  in 
phase  in  such  a  manner  as  to  magnetize  the  field  and  make  up  for 
the  deficiency  in  the  field  excitation.  That  is,  the  armature 
current  becomes  lagging.  Inversely,  if  the  field  excitation  of  the 
synchronous  motor  is  increased,  the  magnetic  flux  still  must 
remain  the  same  as  to  correspond  to  the  impressed  voltage  at 
synchronous  speed,  and  the  armature  current  so  becomes 
demagnetizing — that  is,  leading. 

By  varying  the  field  excitation  of  a  synchronous  motor  or 
converter,  quadrature  components  of  current  can  be  produced 
at  will,  proportional  to  the  variation  of  the  field  excitation  from 
the  value  that  gives  a  magnetic  flux,  which  at  synchronous  speed 
just  consumes  the  impressed  voltage  (after  allowing  for  the 
impedance  of  the  motor). 

Phase  control  of  transmission  lines  is  especially  suited  for 
circuits  supplying  synchronous  motors  or  converters;  since  such 
machines,  in  addition  to  their  mechanical  or  electrical  load, 
can  with  a  moderate  increase  of  capacity  carry  or  produce  con- 
siderable values  of  wattless  current.  For  instance,  a  quadrature 
component  of  current  equal  to  50  per  cent,  of  the  power  com- 
ponent of  current  consumed  by  a  synchronous  motor  would 
increase  the  total  current  only  to  Vl  -f  0.52  =  1.118,  or  11.8 
per  cent.,  while  a  quadrature  component  of  current  equal  to  30 
per  cent,  of  the  power  component  of  the  current  would  give  an 


PHASE  CONTROL  99 

increase  of  4.4  per  cent,  only,  that  is,  could  be  carried  by  the 
motor  armature  without  any  appreciable  increase  of  the  motor 
heating. 

Phase  control  depends  upon  the  inductive  reactance  of  the 
line  or  circuit  between  generating  and  receiving  voltage,  e0  and 
e,  and  where  the  inductive  reactance  of  the  transmission  line 
is  not  sufficient,  additional  reactance  may  be  inserted  in  the 
form  of  reactive  coils  or  high  internal  reactance  transformers. 
This  is  usually  the  case  in  railway  transmissions  to  synchronous 
converters.  Phase  control  is  extensively  used  for  voltage 
control  in  railway  power  transmission  by  compounded  syn- 
chronous converters.  It  is  also  used  to  a  considerable  extent 
in  very  long  distance  transmission,  for  controlling  the  voltage 
and  the  power-factor;  in  a  distribution  system  for  controlling 
the  power-factor  of  the  system. 

While,  therefore,  the  resistance,  r,  of  the  line  is  fixed,  as  it 
would  not  be  economical  to  increase  it,  the  reactance,  x,  can  be 
increased  beyond  that  given  by  line  and  transformer,  by  the 
insertion  of  reactive  coils,  and  therefore  can  be  adjusted  so  as 
to  give  best  results  in  phase  control,  which  are  usually  obtained 
when  the  quadrature  component  of  the  current  is  a  minimum. 

82.  Let,  then, 

e  =  voltage  at  receiving  circuit,  chosen  as  zero  vector. 

I  =  i—  jif  =  current  in  receiving  circuit,  comprising  a  power 
component,  i}  which  depends  upon  the  load  in  the  receiving 
circuit,  and  a  quadrature  component,  i't  which  can  be  varied  to 
suit  the  requirements  of  regulation,  and  is  considered  positive 
when  lagging,  negative  when  leading. 

EQ  =  e'o  —  jeQ"  =  voltage  impressed  upon  the  system  at  the 
generator  end,  or  supply  voltage,  and  the  absolute  value  is 


e,  =       eV  +  e'V. 

Z  =  r  +  jx  =  impedance  of  the  circuit  between  voltage  e 
and  voltage  e0)  and  the  absolute  value  is  z  =  V  r2  +  x2. 

If  e  =  terminal  voltage  of  receiving  station,  e0  =  terminal 
voltage  of  generating  station,  Z  —  impedance  of  transmission 
line;  if  e'=  nominal  induced  e.m.f.  of  receiving  synchronous 
machine,  that  is,  voltage  corresponding  to  its  field  excitation, 
and  60  =  nominal  induced  e.m.f.  of  generator,  Z  also  includes 
the  synchronous  impedance  of  both  machines,  and  of  step-up  and 
step-down  transformers,  where  used, 


100         ALTERNATING-CURRENT  PHENOMENA 

It  is 

E0  =  e  +  ZI, 

or, 

Eo  =  (e  +  ri  +  xif)  -  j(ri'  -  xi),  (i) 

and  in  absolute  value  we  have 

602  =  (e  +  ri  +  xi'Y  +  (rif  -  xi}\  (2) 

This  is  the  fundamental  equation  of  phase  control,  giving 
the  relation  of  the  two  voltages,  e  and  e0,  with  the  two  com- 
ponents of  current,  i  and  i't  and  the  circuit  constants  r  and  x. 

From  equation  (2),  follows: 

e  =  VV  -  (rif  -  xi)*  -  (ri  +  xi'),  (3) 

expressing  the  receiver  voltage,  e,  as  a  function  of  e$  and  I. 

And:  

.,  /eo2 fer         A2       ex  ,A, 

1  =±V^-(?  +  v  -.?• 

Denoting 

tan  0  =  -»  (5) 

where  0  is  the  phase  angle  of  the  line  impedance,  we  have 

r  =  z  cos  6  and  x  =  z  sin  6  (6) 

and 

..  fe02       /c  cos  0   ,     A 2       e  sin  0  ,-v 

1     -v?-(-r+i)     — 

gives  the  reactive  component  of  the  current,  ift  required  by  the 
power  component  of  the  current,  i,  at  the  voltages,  e  and  e0. 
83.  The  phase  angle  of  the  impressed  e.m.f.,  E0,  is,  from  (1), 

tan  00  =  — :  — ^-T- — -,'  (8) 

e  +  n  +  zi 

the  phase  angle  of  the  current 

tan  0i  =  ^>  (9) 

hence,  to  bring  the  current,  7,  into  phase  with  the  impressed 
e.m.f.,  EQ,  or  produce  unity  power-factor  at  the  generator  ter- 
minal, eo,  it  must  be 

$o  =  0i J 
hence, 


e  +  ri 


PHASE  CONTROL 


and  herefrom  follows 


2x 


101 


(10) 


4  & 

hence  always  negative,  or  leading*,  but  i'  —  0  for  i  =  0,  or  at 
no-load. 

From  equation  (10)  follows  that  i'  becomes  imaginary,  if 
the  term  under  the  square  root,  (e2  —  4  x2i'2),  becomes  negative, 
that  is,  if 


that  is,  the  maximum  load,  or  power  component  of  current, 
at  which  unity  power-factor  can  still  be  maintained  at  the 
supply  voltage,  eQ,  is  given  by 

e 


2000 


200      400 


800      1000     1-200     1400     1600     1300    2000 
AMPERES  LOAD  *£ 

FIG.  77.' 


(11) 


and  the  leading  quadrature  component  of  current  required  to 
compensate  for  the  line  reactance  x  at  maximum  current,  im,  is 
from  equation  (10), 

im'  =  ~'  (12) 

that  is,  in  this  case  of  the  maximum  load  which  can  be  delivered 
at  e,  with  unity  power-factor  at  eQ,  the  total  current,  /,  leads 
the  receiver  voltage,  e,  by  45°. 


102         ALTERNA  TING-C URRENT  PHENOMENA 

Substituting  the  value,  i',  of  equation  (10),  which  compensates 
for  the  line  reactance,  x,  and  so  gives  unity  power-factor  at  60, 
into  equation  (2),  gives  as  required  supply  voltage  CQ. 

e*z*    ,    (x-r)  (e-2 


As  illustration  are  shown,  in  Fig.  77,  with  the  load  current,  if 
as  abscissas,  the  values  of  leading  quadrature  component  of 
current,  if,  and  of  generator  voltage,  e0,  for  the  constants 
6  =  400  volts;  r  =  0.05  ohm,  and  x  =  0.10  ohm. 

84.  More  frequently  than  for  controlling  the  power-factor, 
phase  control  is  used  for  controlling  the  voltage,  that  is,  to 
maintain  the  receiver  voltage,  e,  constant,  or  raise  it  with  in- 
creasing load,  if  at  constant  generator  voltage,  eQ. 

In  this  case,  equation  (4)  gives  the  quadrature  component 
of  current,  ir,  required  by  current,  i,  at  constant  receiver  vol- 
tage, c,  and  constant  generator  voltage,  eQ. 

Since  the  equation  (4)  of  i'  contains  a  square  root,  the  maxi- 
mum value  of  it  that  is,  the  maximum  load  which  can  be  carried 
at  constant  voltage,  e  and  CQ,  is  given  by  equating  the  term  under 
the  square  root  to  zero 


as 

t-m  = 


and  the  corresponding  quadrature  component  of  current,  by 
(4),  is 

.  ex  esin0 


that  is,  leading. 

From  equation  (14)  follows  as  the  impedance,  2,  which,  at 
constant  line-resistance,  r,  gives  the  maximum  value  of  im 


2w  =  2r-»  (16) 

€Q 

and  for  this  value  of  impedance,  2m,  substituting  in  (14)  and  (15) 
^-,       and       ."--  (17) 


PHASE  CONTROL  103 

The  maximum  load,  i,  which  can  be  delivered  at  constant 
voltage,  e,  therefore  depends  upon  the  line  impedance,  and  the 
voltages,  e  and  eQ. 

Since  eQ  and  e  are  not  very  different  from  each  other,  the  ratio 

ft 

-  in  equation  (16)  is  approximately  unity,  and  the  impedance, 

Co 

2,  which  permits  maximum  load  to  be  transmitted,  is  approxi- 
mately twice  the  line  resistance,  r,  or  rather  slightly  less. 

z  <  2r, 
gives 

x  < 


A  relatively  low  line-reactance,  x,  so  gives  maximum,  output. 
In  practice,  a  far  higher  reactance,  x,  is  used,  since  it  gives 
sufficient  output  and  a  lesser  quadrature  component  of  current. 

By  substituting  i  =  0  in  equation  (4),  the  value  of  the  quad- 
rature component  of  current  at  no-load  is  found  as 


.,       Veo222  —  eV  —  ex 
i  o  =  -3 


V  602  —  e2  cos2  0  —  e  sin  0 


z 
This  can  be  written  in  the  form 


(18) 


..        V(e02  -  e2)  +  e2  sin2  8  —  e  sin  6 

~T 

and  then  shows  that  for  e  =  e0,  i'Q  =  0,  or  no  quadrature  com- 
ponent of  current  exists  at  no-load;  for  e  >  e0,  i'Q  <  0  or  nega- 
tive, that  is,  the  quadrature  component  of  current  is  already 
leading  at  no-load.  For:  e  <  e0,  t'o  >  0  or  lagging,  that  is,  the 
quadrature  component  of  current  i'Q  is  lagging  at  no-load,  be- 
comes zero  at  some  load,  and  leading  at  still  higher  loads. 

The  latter  arrangement,  e  <  e0,  is  generally  used,  as  the  quad- 
rature component  of  current  passes  through  zero  at  some  inter- 
mediate load,  and  so  is  less  over  the  range  of  required  load  than 
it  would  be  if  z'0  were  0  or  negative. 

From  (18)  follows  that  the  larger  2,  and  at  constant  resistance 
r,  also  JE,  the  smaller  the  quadrature  component  of  current. 
That  is,  increase  of  the  line  reactance,  x,  reduces  the  quadrature 
current  at  no-load,  i'0,  and  in  the  same  way  at  load,  that  is,  im- 
proves the  power-factor  of  the  circuit,  and  so  is  desirable,  and  the 
insertion  of  reactive  coils  in  the  line  for  this  reason  customary. 


104         ALTERNATING-CURRENT  PHENOMENA 

Increase  of  reactance,  however,  reduces  the  maximum  output 
im,  and  too  large  a  reactance  is  for  this  reason  objectionable. 
Let 

i  =  ii 

be  the  load  at  which  the  quadrature  component  of  current 
vanishes,  i'  =  0,  that  is,  the  receiver  circuit  has  unity  power- 
factor. 
Substituting  i  =  ii,  i'  =  0  into  equation  (2)  gives 

eo2  =  (e  +  rii)8  +  xV  (19) 

and,  substituting  (19)  in  (4),  (18),  (14),  gives 
reactive   component   of   current 


.,         /e2sin20       2  6  cos  0..                                     -e  sin  0      ,0ft. 
*'  =  >J      g2       + (*i  -  0  +  (*i2  -  *2) — '     (20) 


and  at  no-load 


e2  sin2  0 
V~^~ 


+  2^.cos0  +  .i2  _  ^  (21) 


Maximum  output  current 


it/-          A  oti  OOS  0  6  COS  0  /oo\ 

^  =  \fa  H :     -  +  *i2  -  -     —  (22) 


85.  Of  importance  in  phase  control  for  constant  voltage,  e, 
at  constant  eQ,  are  the  three  currents 

ii,  the  power  component  of  current  at  which  the  quadra- 
ture component  of  current  vanishes:  i'  =  0. 
im,  the  maximum  load  which  can  be  transmitted  at  con- 
stant voltage,  e. 

i'o,  the  reactive  component  of  current  at  no-load. 
The  equation  of  phase  control,  (2),  however,  contains  only  two 
quantities  which  can  be  chosen:  The  reactance,  x,  which  can 
be  increased  by  inserting  reactive  coils,  and  the  generator  vol- 
tage, e0,  which  can  be  made  anything  desired,  even  with  an 
existing  generating  station,  since  between  e  and  e0  practically 
always  transformers  are  interposed,  and  their  ratio  can  be 
chosen  so  as  to  correspond  to  any  desired  generator  voltage,  eQ, 
as  they  usually  are  supplied  with  several  voltage  steps. 

Of  the  three  quantities,  ii,  im  and  i'o,  only  two  can  be  chosen, 
and  the  constants,  x  and  eQ,  derived  therefrom.  The  third 
current  then  also  follows,  and  if  the  value  found  for  it  does  not 
suit  the  requirements  of  the  problem,  other  values  have  to  be 
tried.  For  instance,  choosing  i\  as  corresponding  to  three-fourths 


PHASE   CONTROL  105 

load,  and  i'Q  fairly  small,  gives  very  good  power-factors  over  the 
whole  range  of  load,  but  a  relatively  low  value  of  im,  and  where 
very  great  overload  capacities  are  required,  im  may  not  be 
sufficient,  and  ii  may  have  to  be  chosen  corresponding  to  full-load 
and  a  higher  value  of  i'0  permitted,  that  is,  some  sacrifice  made 
in  the  power-factor,  in  favor  of  overload  capacity. 
So,  for  instance,  the  values  may  be  chosen 

iij  corresponding  to  full-load, 
and  required  that  i'Q  does  not  exceed  half  of  full-load  current; 

and  that  the  synchronous  converter  or  motor  can  carry  at  least 
100  per  cent,  overload,  that  is, 

im  >  2  ii. 

We  then  can  put,  im  —  2  ii  c  and  i'o  =  —  — ,  (23) 

C 

and  substitute  (23)  in  (19),  (22)  and  determine  x,  e0,  c. 

86.  The  variation  of  the  reactive  current,  i'  with  the  load, 
I,  equation  (4),  is  brought  about  by  varying  the  field  excitation 
of  the  receiving  synchronous  machine.  Where  the  load  on  the 
synchronous  machine  is  direct-current  output,  as  in  a  motor 
generator  and  especially  a  converter,  the  most  convenient  way 
of  varying  the  field  excitation  with  the  load  is  automatically, 
by  a  series  field-coil  traversed  by  the  direct-current  output. 
The  field  windings  of  converters  intended  for  phase  control — 
as  for  the  supply  of  power  to  electric  railways,  from  substations 
fed  by  a  high-potential  alternating-current  transmission  line — 
are  compound-wound,  and  the  shunt  field  is  adjusted  for  under- 
excitation,  so  as  to  produce  at  no-load  the  lagging  current,  i'Q, 
and  the  series  field  adjusted  so  as  to  make  the  reactive  compo- 
nent of  current,  i',  disappear  at  the  desired  load,  i\. 

In  this  case,  however,  the  variation  of  the  field  excitation  by 
the  series  field  is  directly  proportional  to  the  load,  as  is  also  the 
variation  of  if,  that  is,  it  varies  from  if  —  i'Q  for  i  =  0,  to  i'  —  0 
for  i  =  ii,  and  can  be  expressed  by  the  equation 

(24) 
(25) 


=  q(ii  -  i) 
where 


106         ALTERNATING-CURRENT  PHENOMENA 

is  the  ratio  of  (reactive)  no-load  current,  z'o,  to  (effective)  non- 
inductive  load  current,  i\. 

To  maintain  constant  voltage,  e,  at  constant,  eQ,  the  required 
variation  of  i'  is  not  quite  linear,  and  with  a  linear  variation  of 
i',  as  given  by  a  compound  field-winding  on  the  synchronous 
machine,  the  receiver-voltage,  e,  at  constant  impressed  voltage 
does  not  remain  perfectly  constant,  but  when  adjusted  for  the 
same  value  at  no-load  and  at  full-load,  e  is  slightly  high  at  inter- 
mediate loads,  low  at  higher  loads.  It  is,  however,  sufficiently 
constant  for  all  practical  purposes. 

Choosing  then  the  full-load  current,  ilt  and  the  no-load  current, 
i'o  =  qii,  and  let  the  reactive  component  of  current,  i',  by  a 
compound  field-winding  vary  as  a  linear  function  of  the  load,  i: 


Then,  substituting  ii  and  i'0  =  qii  in  the  equations  (2)  for 
phase  control: 

No-load:  i  =  0,        i'  =  qii; 

eo2  =  (e  +  qxii)2  +  qri^.  (26) 

Full  load:  ii  =  i1}        i'  =  0; 

eo2  =  (e  +  rii)*  +  xiS.  (27) 

From  these  equations  (26)  and  (27)  then  calculate  the  required 
reactance,  x,  and  the  generator  voltage,  e0,  as: 


2fj.  Jr. 


and  from  (27)  or  (26)  the  voltage,  eQ. 

The  terminal  voltage  at  the  receiving  circuit  then  is,  by  equa- 
tion (3) : 

e  =  ^/eQ2-[qrii-(qr+x)i]2  -  ((r  -  qx)i  +  qxii).      (29) 

As  an  example  is  shown,  in  Fig.  78,  the  curve  of  receiving 
voltage,  e,  with  the  load,  i,  as  abscissas,  for  the  values: 

e  =  400  volts  at  no-load  and  at  full-load, 
ii  =  500  amp.  at  full-load,  power  component  of  current, 
i'o  =  200  amp.,  lagging  reactive  or  quadrature  component 
of  current  at  no-load, 

hence    q  =  0.4, 

i'  =  200  -  0.4  », 

and       r  =  0.05  ohm. 


PHASE  CONTROL 


107 


From  equation  (28)  then  follows: 

x  =  0.381  ±  0.165  ohm. 
Choosing  the  lower  value: 

x  =  0.216  ohm. 
gives,  from  equation  (27) : 

e0  =  443.4  volts; 
hence 


e  =  \  196,420+  5.76  i  -0.0576  z2  -  (43.2  -  0.0264  j). 

For  comparison  is  shown,  in  Fig.  78,  the  receiving  voltage,  e', 
at  the  same  supply  voltage,  eQ  =  443.4  volts,  but  without  phase 
control,  that  is,  with  a  non-inductive  receiver-circuit. 


800 


•100 


300 


1000 


FIG.  78. 

87.  Equation  (28)  shows  that  there  are  two  values  of  x: 
Xi  and  x2;  and  corresponding  thereto  two  values  of  e0:e0i  and  e02, 
which  as  constant-supply  voltage  give  the  same  receiver-voltage, 
e,  at  no-load  and  at  full-load,  and  so  approximately  constant 
receiver-voltage  throughout. 

One  of  the  two  reactances,  X2,  is  much  larger  than  the  other, 
Xi,  and  the  corresponding  voltage,  e02,  accordingly  larger  than  eQi. 

In  addition  to  the  terminal  voltage,  e,  at  the  receiver-circuit, 
there  are  therefore  two  further  points  of  constant  voltage  in  the 
system:  eoi,  distant  from  e  by  the  resistance,  r,  and  reactance, 
xif  and  :  e^,  distant  from  eoi  by  the  reactance  XQ  =  x%  —  xi. 


108         ALTERNATING-CURRENT  PHENOMENA 

That  is,  by  the  proper  choice  of  the  reactances,  Xi  and  z0, 
three  points  of  the  system  can  be  maintained  automatically  at 
approximately  constant  voltage,  by  phase  control:  e,  e0i  and  602- 

Such  multiple-phase  control  can  advantageously  be  employed 
by  using: 

e  as  the  terminal  voltage  of  the  receiving  circuit, 

601  as  the  generator  terminal  voltage  e0,  and 

602  as  the  nominal  induced  e.m.f.  of  the  generator,  that  is,  the 
voltage  corresponding  to  the  field-excitation.     Constancy  of  eQZ 
accordingly  means  constant  field-excitation. 

That  is,  with  constant  field-excitation  of  the  generator,  the 
voltage  remains  approximately  constant,  by  multiple-phase  con- 
trol, at  the  generator  busbars  as  well  as  at  the  terminals  of 
the  receiving  circuit,  at  the  end  of  the  transmission  line  of 
resistance,  r. 

In  this  case: 

Xi  =  reactance  of  transmission  line  plus  reactive  coils  inserted 
in  the  line  (usually  at  the  receiving  station). 

XQ  =  #2  —  Xi  =  synchronous  reactance  of  the  generator  plus 
reactive  coils  inserted  between  generator  and  generator  bus- 
bars, where  necessary. 

Since  the  generator  also  contains  a  small  resistance,  7*0,  the 
two  values  of  reactance,  x\  and  #2  =  x\  +  XQ,  are  given  by  the 
equation  (28)  as: 


1-g* 
and 


Assuming  in  above  example: 

T-Q  =  0.01  ohm 
gives 

x2  =  0.440  ohm; 
hence, 

XQ  =  0.224  ohm. 

The  curve  of  nominal  generated  e.m.f.,  602,  of  the  generator  is 
shown  in  Fig.  78  as  £02- 


PHASE  CONTROL  109 

That  is,  at  constant  field-excitation,  corresponding  to  a  nomi- 
nal generated  e.m.f., 

e02  =  488.2  volts. 
The  generator  of  synchronous  impedance, 

Z0  =  r0  4-  jx0  =  0.01  -f  0.224  j  ohms, 

maintains  approximately  constant  voltage  at  its  own  terminals, 
or  at  the  generator  busbars, 

e0  =  443.4  volts, 
and  at  the  same  time  maintains  constant  voltage, 

e  =  400  volts, 
at  the  end  of  a  transmission  line  of  impedance, 

Z  =  r  +  jxi  =  0.06  +  0.216  j  ohms, 

if  by  phase  control  in  the  receiving  circuit,  by  compounded 
converter,  the  reactive  or  quadrature  component  of  current, 
i',  is  varied  with  the  load  or  power  component  of  current,  i, 
and  proportional  thereto,  that  is: 

i'  =  ?0'i  -  i) 
=  200  -  0.4  i. 

88.  To  adjust  a  circuit  experimentally  for  phase  control 
for  constant  voltage,  by  overcomppunded  synchronous  converter : 
at  constant-supply  voltage  and  no-load  on  the  converter — with 
the  transmission  line  with  its  transformers,  reactances,  etc., 
or  an  impedance  equal  thereto,  in  the  circuit  between  con- 
verter and  supply  voltage — the  shunt  field  of  the  converter  is 
adjusted  by  the  field  rheostat  so  as  to  give  the  desired  direct- 
current  voltage  at  the  converter  brushes.  Then  load  is  put  on 
the  converter,  and,  without  changing  the  supply  voltage  or  the 
adjustment  of  the  shunt  field,  the  rheostat  or  shunt  across  the 
series  field  of  the  converter  is  adjusted  so  as  to  give  the  desired  di- 
rect-current voltage. 

If  the  supply  voltage  can  be  varied,  as  is  usually  provided 
for  by  different  voltage  taps  on  the  transformer,  then,  before 
adjusting  the  converter  fields  as  described  above,  first  the  proper 
supply  voltage  is  found.  This  is  done  by  loading  the  converter 
with  the  current,  at  which  unity  power-factor  at  the  converter  is 


110         ALTERNATING-CURRENT  PHENOMENA 

desired — for  instant  full-load — and  then  varying  the  converter 
shunt  field  so  as  to  get  minimum  alternating-current  input,  and 
varying  the  supply  voltage  so  as  to  get — at  minimum  alternat- 
ing-current input — the  desired  direct-current  voltage.  Where 
the  supply  voltage  can  only  be  varied  in  definite  steps:  at  some 
voltage  step,  the  converter  field — at  the  desired  non-inductive 
load — is  adjusted  for  minimum  alternating-current  input; 
if  then  the  direct-current  voltage  is  too  low,  the  transformer 
connections  are  changed  to  the  next  higher  supply  voltage 
step;  if  the  direct-current  voltage  is  too  high,  the  change  is 
made  to  the  next  lower  supply  voltage  step,  until  that  supply 
voltage  step  is  found,  which,  at  the  adjustment  of  the  converter 
field  for  minimum  alternating-current  input,  brings  the  direct- 
current  voltage  nearest  to  that  desired.  Then  for  this  supply 
voltage  step,  the  converter  field  circuits  are  adjusted  for  phase 
control,  as  above  described. 


SECTION  III 

POWER  AND  EFFECTIVE 
CONSTANTS 


CHAPTER  XII 
EFFECTIVE  RESISTANCE  AND  REACTANCE 

89.  The  resistance  of  an  electric  circuit  is  determined : 

1.  By  direct  comparison  with  a  known  resistance  (Wheat- 
stone  bridge  method,  etc.). 

This  method  gives  what  may  be  called  the  true  ohmic  resist- 
ance of  the  circuit. 

2.  By  the  ratio : 

Volts  consumed  in  circuit 

Amperes  in  circuit 

In  an  alternating-current  circuit,  this  method  gives,  not  the 
resistance  of  the  circuit,  but  the  impedance, 
z  =  -y/r2  +  x2. 

3.  By  the  ratio: 

Power  consumed 

(Current)2 

where,  however,  the  "power"  does  not  include  the  work  done 
by  the  circuit,  and  the  counter  e.m.fs.  representing  it,  as,  for 
instance,  in  the  case  of  the  counter  e.m.f.  of  a  motor. 

In  alternating-current  circuits,  this  value  of  resistance  is  the 
power  coefficient  of  the  e.m.f., 

Power  component  of  e.m.f. 

Total  current 

It  is  called  the  effective  resistance  of  the  circuit,  since  it  represents 
the  effect,  or  power,  expended  by  the  circuit.  The  power  coeffi- 
cient of  current, 

Power  component  of  current 

Total  e.m.f. 

is  called  the  effective  conductance  of  the  circuit. 

Ill 


112         ALTERNATING-CURRENT  PHENOMENA 

In  the  same  way,  the  value, 

Wattless  component  of  e.m.f. 
Total  current 

is  the  effective  reactance,  and  > 

Wattless  component  of  current 

0    =    rrTT — i f > 

Total  e.m.f. 

is  the  effective  susceptance  of  the  circuit. 

While  the  true  ohmic  resistance  represents  the  expenditure 
of  power  as  heat  inside  of  the  electric  conductor  by  a  current 
of  uniform  density,  the  effective  resistance  represents  the  total 
expenditure  of  power. 

Since  in  an  alternating-current  circuit,  in  general  power  is 
expended  not  only  in  the  conductor,  but  also  outside  of  it, 
through  hysteresis,  secondary  currents,  etc.,  the  effective  resist- 
ance frequently  differs  from  the  true  ohmic  resistance  in  such 
way  as  to  represent  a  larger  expenditure  of  power. 

In  dealing  with  alternating-current  circuits,  it  is  necessary, 
therefore,  to  substitute  everywhere  the  values  "  effective  re- 
sistance," " effective  reactance/'  "effective  conductance/'  and 
"  effective  susceptance,"  to  make  the  calculation  applicable  to 
general  alternating-current  circuits,  such  as  inductive  reactances 
containing  iron,  etc. 

While  the  true  ohmic  resistance  is  a  constant  of  the  circuit, 
depending  only  upon  the  temperature,  but  not  upon  the  e.m.f., 
etc.,  the  effective  resistance  and  effective  reactance  are,  in  gen- 
eral, not  constants,  but  depend  upon  the  e.m.f.,  current,  etc. 
This  dependence  is  the  cause  of  most  of  the  difficulties  met  in 
dealing  analytically  with  alternating-current  circuits  containing 
iron. 

90.  The  foremost  sources  of  energy  loss  in  alternating-current 
circuits,  outside  of  the  true  ohmic  resistance  loss,  are  as  follows: 

1.  Molecular  friction,  as, 
(a)  Magnetic  hysteresis; 
(6)   Dielectric  hysteresis. 

2.  Primary  electric  currents,  as, 

(a)  Leakage  or  escape  of  current  through  the  insulation, 

brush  discharge,  corona. 

(b)  Eddy  currents  in  the  conductor  or  unequal  current 

distribution. 


EFFECTIVE  RESISTANCE  AND  REACTANCE    113 

3.  Secondary  or  induced  currents,  as, 

(a)  Eddy  or  Foucault  currents   in  surrounding  magnetic 

materials; 
.  (b)  Eddy  or  Foucault  currents  in  surrounding  conducting 

materials  ; 
(c)  Secondary  currents  of  mutual  inductance  in  neighboring 

circuits. 

4.  Induced  electric  charges,  electrostatic  induction  or  influence. 

While  all  these  losses  can  be  included  in  the  terms  effective 
resistance,  etc.,  the  magnetic  hysteresis  and  the  eddy  currents 
are  the  most  frequent  and  important  sources  of  energy  loss. 

Magnetic  Hysteresis 

91.  In  an  alternating-current  circuit  surrounded  by  iron  or 
other  magnetic  material,  energy  is  expended  outside  of  the  con- 
ductor in  the  iron,  by  a  kind  of  molecular  friction,  which,  when 
the  energy  is  supplied  electrically,  appears  as  magnetic  hysteresis, 
and  is  caused  by  the  cyclic  reversals  of  magnetic  flux  in  the  iron 
in  the  alternating  magnetic  field. 

To  examine  this  phenomenon,  first  a  circuit  may  be  con- 
sidered, of  very  high  inductive  reactance,  but  negligible  true 
ohrnic  resistance;  that  is,  a  circuit  entirely  surrounded  by  iron, 
as,  for  instance,  the  primary  circuit  of  an  alternating-current 
transformer  with  open  secondary  circuit. 

The  wave  of  current  produces  in  the  iron  an  alternating  mag- 
netic flux  which  generates  in  the  electric  circuit  an  e.m.f. — the 
counter  e.m.f.  of  self-induction.  If  the  ohmic  resistance  is 
negligible,  that  is,  practically  no  e.m.f.  consumed  by  the  resist- 
ance, all  the  impressed  e.m.f.  must  be  consumed  by  the  counter 
e.m.f.  of  self-induction,  that  is,  the  counter  e.m.f.  equals  the 
impressed  e.m.f.;  hence,  if  the  impressed  e.m.f.  is  a  sine  wave, 
the  counter  e.m.f.,  and,  therefore,  the  magnetic  flux  which 
generates  the  counter  e.m.f.,  must  follow  a  sine  wave  also.  The 
alternating  wave  of  current  is  not  a  sine  wave  in  this  case,  but  is 
distorted  by  hysteresis.  It  is  possible,  however,  to  plot  the  cur- 
rent wave  in  this  case  from  the  hysteretic  cycle  of  magnetic  flux. 

From  the  number  of  turns,  n,  of  the  electric  circuit,  the  effective 
counter  e.m.f.,  E,  and  the  frequency,  /,  of  the  current,  the  maxi- 
mum magnetic  flux,  $,  is  found  by  the  formula: 

E  =  VZirnf®  10-8; 


114         ALTERNATING-CURRENT  PHENOMENA 
hence, 


A  maximum  flux,  <J>,  and  magnetic  cross-section,  A,  give  the 
maximum  magnetic  induction,  B  =  -r- 

A. 

If  the  magnetic  induction  varies  periodically  between  +  B 
and  —  B,  the  magnetizing  force  varies  between  the  corresponding 
values  +  /  and  —  /,  and  describes  a  looped  curve,  the  cycle  of 
hysteresis. 

If  the  ordinates  are  given  in  lines  of  magnetic  force,  the 
abscissas  in  tens  of  ampere-turns,  then  the  area  of  the  loop 
equals  the  energy  consumed  by  hysteresis  in  ergs  per  cycle. 

From  the  hysteretic  loop  the  instantaneous  value  of  magnetiz- 
ing force  is  found,  corresponding  to  an  instantaneous  value  of 
magnetic  flux,  that  is,  of  generated  e.m.f.;  and  from  the  mag- 
netizing force,  /,  in  ampere-turns  per  units  length  of  magnetic 
circuit,  the  length,  /,  of  the  magnetic  circuit,  and  the  number 
of  turns,  n,  of  the  electric  circuit,  are  found  the  instantaneous 
values  of  current,  i,  corresponding  to  a  magnetizing  force,  /, 
that  is,  magnetic  induction,  B,  and  thus  generated  e.m.f.,  e,  as: 


92.  In  Fig.  79,  four  magnetic  cycles  are  plotted,  with  maximum 
.values  of  magnetic  induction,  B  =  2,000,  6,000,  10,000,  and  16,- 
000,  and  corresponding  maximum  magnetizing  forces,  /  =  1.8, 
2.8,  4.3,  20.0.  They  show  the  well-known  hysteretic  loop,  which 
becomes  pointed  when  magnetic  saturation  is  approached. 

These  magnetic  cycles  correspond  to  sheet  iron  or  sheet  steel, 
of  a  hysteretic  coefficient,  rj  =  0.0033,  and  are  given  with 
ampere-turns  per  centimeter  as  abscissas,  and  kilolines  of  mag- 
netic force  as  ordinates. 

In  Figs.  80  and  81,  the  curve  of  magnetic  induction  as  derived 
from  the  generated  e.m.f.  is  a  sine  wave.  For  the  different  values 
of  magnetic  induction  of  this  sine  curve,  the  corresponding  values 
of  magnetizing  force  /,  hence  of  current,  are  taken  from  Fig.  79, 
and  plotted,  giving  thus  the  exciting  current  required  to  produce 
the  sine  wave  of  magnetism;  that  is,  the  wave  of  current  which 
a  sine  wave  of  impressed  e.m.f.  will  establish  in  the  circuit. 


EFFECTIVE  RESISTANCE  AND  REACTANCE    115 


FIG.  79. — Hysteretic  cycle  of  sheet  iron. 


I 


\ 


^ 


\ 


\ 


B  =  2000 
/=1.8 
1=  1.8 


B  =  6000 
/=2.8 
1  =  2.9 


40 


FIG.  80. 


116 


ALTERNATING-CURRENT  PHENOMENA 


As  shown  in  Figs.  80  and  81,  these  waves  of  alternating  current 
are  not  sine  waves,  but  are  distorted  by  the  super-position  of 
higher  harmonics,  and  are  complex  harmonic  waves.  They 
reach  their  maximum  value  at  the  same  time  with  the  maximum  of 
magnetism,  that  is,  90°  ahead  of  the  maximum  generated  e.m.f., 
and  hence  about  90°  behind  the  maximum  impressed  e.m.f.,  but 
pass  the  zero  line  considerably  ahead  of  the  zero  value  of  magne- 
tism, of  42°,  52°,  50°  and  41°,  respectively. 


FIG.  81. 

The  general  character  of  these  current  waves  is,  that  the  maxi- 
mum point  of  the  wave  coincides  in  time  with  the  maximum 
point  of  the  sine  wave  of  magnetism;  but  the  current  wave  is 
bulged  out  greatly  at  the  rising,  and  hollowed  in  at  the  decreasing, 
side.  With  increasing  magnetization,  the  maximum  of  the  cur- 
rent wave  becomes  more  pointed,  as  shown  by  the  curves  of 
Fig.  81,  for  B  =  10,000;  and  at  still  higher  saturation  a  peak  is 


EFFECTIVE  RESISTANCE  AND  REACTANCE    117 

formed  at  the  maximum  point,  as  in  the  curve  for  B  =  16,000. 
This  is  the  case  when  the  curve  of  magnetization  reaches  within 
the  range  of  magnetic  saturation,  since  in  the  proximity  of 
saturation  the  current  near  the  maximum  point  of  magnetization 
has  to  rise  abnormally  to  cause  even  a  small  increase  of  magneti- 
zation. The  four  curves,  Figs.  80  and  81  are  not  drawn  to  the 
same  scale.  The  maximum  values  of  magnetizing  force,  corre- 
sponding to  the  maximum  values  of  magnetic  induction,  B  = 
2,000,  6,000,  10,000,  and  16,000  lines  of  force  per  square  centi- 
meter, are/  =  1.8,  2.8,  4.3,  and  20.0  ampere-turns  per  centimeter. 
In  the  different  diagrams  these  are  represented  in  the  ratio  of 
8:6:4:1,  in  order  to  bring  the  current  curves  to  approximately 
the  same  height.  The  magnetizing  force,  in  c.g.s.  units,  is 

H  =  47T/10/  =  1.257/. 

93.  The  distortion  of  the  current  waves,  /,  in  Figs.  80  and  81, 
is  almost  entirely  due  to  the  magnetizing  current,  and  is  caused 
by  the  disproportionality  between  magnetic  induction,  B,  and 
magnetizing  force,  /,  as  exhibited  by  the  magnetic  characteristic 
or  saturation  curve,  and  is  very  little  due  to  hysteresis. 

Resolving  these  curves,  /,  of  Figs.  80  and  81  into  two  com- 
ponents, one  in  phase  with  the  magnetic  induction,  J5,  or  sym- 
metrical thereto,  hence  in  quadrature  with  the  induced  e.m.f., 
and  therefore  wattless:  the  magnetizing  current,  im;  and  the 
other,  in  time  quadrature  with  the  magnetic  induction,  B,  hence 
in  phase,  or  symmetrical,  with  the  generated  e.m.f.,  that  is, 
representing  power:  the  hysteresis  power-current,  fa.  Then  we 
see  that  the  hysteresis  power-current,  fa}  is  practically  a  sine 
wave,  while  the  magnetizing  current,  im,  differs  considerably 
from  a  sine  wave,  and  tends  toward  peakedness — the  more  the 
higher  the  magnetic  induction,  J5,  that  is,  the  more  magnetic 
saturation  is  approached,  so  that  for  B  =  16,000  a  very  high 
peak  is  shown,  and  the  wave  of  magnetizing  current,  im,  does 
not  resemble  a  sine  wave  at  all,  but  at  the  maximum  value  is 
nearly  four  times  higher  than  a  sine  wave  of  the  same  instan- 
taneous values  near  zero  induction  would  have. 

These  curves  of  hysteresis  power-current,  fat  and  magnetiz- 
ing current,  im,  derived  by  resolving  the  distorted  current 
curves,  /,  of  Figs.  80  and  81,  are  plotted  in  Fig.  82,  the  last  one, 
corresponding  to  B  =  16,000,  with  one-quarter  the  ordinates  of 
the  first  three. 


118 


ALTERNATING-CURRENT  PHENOMENA 


As  curves,  symmetrical  with  regard  to  the  maximum  value  of 
B  —  im  — ,  and  to  the  zero  value  of  B  —  4  — ,  these  curves  are 
constructed  thus: 

Let 

&  =  B  sin  0  =  sine  wave  of  magnetic  induction, 


2.0 
1.0 
0 
-1.0 
-2.0 
2.0 
1  0 

;| 

^ 

lh 

^ 

•** 

Xs- 

^ 

1 

1 

^~ 

-—  ^* 

>> 

«^, 

\ 

^- 

—  - 

(j 

- 

^*-* 

»-• 

-^, 

^1 

"^^ 

^ 

*^** 

•** 

^" 

*** 

•~», 

-^ 

^>. 

^ 

x- 

•^ 

^  —  , 

-^*« 

5< 

"^ 

•  — 

^ 

-•»— 

—  - 

1 

m 

^  —  ' 

*^ 

I 

h 

,/ 

/ 

\ 

—  . 

•^ 

^ 

/ 

x^ 

V 

s 

^ 

0 

-1.0 
*2.0 
2.0 
1.0 

0 

1  0 

f 

s* 

? 

\ 

\, 

^ 

^ 

^* 

s 

s^ 

"^ 

^^ 

s 

^ 

N 

^^ 

/ 

^ 

^» 

;/^ 

"N 

\ 

x 

^ 

/ 

/ 

\ 

\ 

\ 

/ 

s 

1     ' 

—  g 

\ 

/ 

0 

/ 

\ 

*^-, 

~s 

" 

^ 

~^ 

\ 

/ 

\ 

/ 

/ 

N 

\ 

/ 

x 

\ 

\ 

/ 

/ 

\ 

ss 

/ 

s/ 

\ 

^ 

V, 

y 

X 

\ 

X 

/ 

/ 

\ 

\ 

p  n 

/ 

/ 

W 

\ 

\ 

>^ 

> 

/ 

\ 

^ 

19  n 

\ 

/ 

/ 

\ 

^-~ 

.  

^^- 

—  ' 

\ 

/ 

/ 

\ 

10.0 
8.0 
6.0 
4.0 
2.0 
0 
-2.0 
-4.0 
-6.0 
-8.0 
-10.0 

19  n 

/ 

/ 

\ 

/ 

\ 

^, 

s 

/ 

s^^ 

1 

\ 

,^- 

X 

^ 

.-** 

-^r 

^ 

,  — 

-- 

„  —  • 

;==• 

-— 

= 

^•^ 

~ 

"^ 

•^ 

\ 

*^ 

1  — 

*—  » 

3^. 

•^c; 

^; 

"** 

/ 

s 

^ 

s 

HYSTERESIS  POWER  CURRENT 
AND  MAGNETIZING  CURRENT 

B  =  2000      6000,    10000,     16000 
/•=1.8     2.8     4.3       20.0 

\ 

/ 

14  o 

\ 

/ 

-16  0 

\ 

/ 

-18  0 

-20,0 

\^ 

J 

FIG.  82. 


then 


<*    ««(/*  +/-*)• 

That  is,  im  is  the  average  value  of  /  for  an  angle  <£,  and  its 
supplementary  angle  180  —  </>,  4  the  average  value  of  /  for  an 
angle  0  and  its  negative  angle  —  0. 


EFFECTIVE  RESISTANCE  AND  REACTANCE    119 


94.  The  distortion  of  the  wave  of  magnetizing  current  is  as 
large  as  shown  here  only  in  an  iron-closed  magnetic  circuit 
expending  power  by  hysteresis  only,  as  in  an  ironclad  trans- 
former on  open  secondary  circuit.  As  soon  as  the  circuit  ex- 
pends power  in  any  other  way,  as  in  resistance  or  by  mutual 


\ 


\ 


\ 


\ 


FIG.  83. — Distortion  of  current  wave  by  hysteresis. 

inductance,  or  if  an  air-gap  is  introduced  in  the  magnetic  circuit, 
the  distortion  of  the  current  wave  rapidly  decreases  and  practi- 
cally disappears,  and  the  current  becomes  more  sinusoidal. 
That  is,  while  the  distorting  component  remains  the  same,  the 
sinusoidal  component  of  the  current  greatly  increases,  and  ob- 


120         ALTERNATING-CURRENT  PHENOMENA 

scures  the  distortion.  For  example,  in  Fig.  83,  two  waves  are 
shown  corresponding  in  magnetization  to  the  last  curve  of 
Fig.  80,  as  the  one  most  distorted.  The  first  curve  in  Fig.  83 
is  the  current  wave  of  a  transformer  at  0.1  load.  At  higher 
loads  the  distortion  is  correspondingly  still  less,  except  where  the 
magnetic  flux  of  self-induction,  that  is,  flux  passing  between 
primary  and  secondary  and  increasing  in  proportion  to  the  load, 
is  so  large  as  to  reach  saturation,  in  which  case  a  distortion 
appears  again  and  increases  with  increasing  load.  The  second 
curve  of  Fig.  83  is  the  exciting  current  of  a  magnetic  circuit 
containing  an  air-gap  whose  length  equals  J^oo  the  length  of  the 
magnetic  circuit.  These  two  curves  are  drawn  to  one-third  the 
size  of  the  curve  in  Fig.  80.  As  shown,  both  curves  are  practir 
cally  sine  waves.  The  sine  curves  of  magnetic  flux  are  shown 
dotted  as  0. 

96.  The  distorted  wave  of  current  can  be  resolved  into  two 
components:  A  true  sine  wave  of  equal  effective  intensity  and 
equal  power  to  the  distorted  wave,  called  the  equivalent  sine  wave, 
and  a  wattless  higher  harmonic,  consisting  chiefly  of  a  term  of 
triple  frequency. 

In  Figs.  80,  81  and  83  are  shown,  as  /,  the  equivalent  sine 
waves,  and  as  i,  the  difference  between  the  equivalent  sine 
wave  and  the  real  distorted  wave,  which  consists  of  wattless 
complex  higher  harmonics.  The  equivalent  sine  wave  of  m.m.f. 
or  of  current,  in  Figs.  80  and  81,  leads  the  magnetism  in  time 
phase  by  34°,  44°,  38°,  and  15.5°,  respectively.  In  Fig.  83  the 
equivalent  sine  wave  almost  coincides  with  the  distorted  curve, 
and  leads  the  magnetism  by  only  9  degrees. 

It  is  interesting  to  note  that  even  in  the  greatly  distorted 
curves  of  Figs.  80  and  81  the  maximum  value  of  the  equivalent 
sine  wave  is  nearly  the  same  as  the  maximum  value  of  the 
original  distorted  wave  of  m.m.f.,  so  long  as  magnetic  saturation 
is  not  approached,  being  1.8,  2.9,  and  4.2,  respectively,  against 
1.8,  2.8,  and  4.3,  the  maximum  values  of  the  distorted  curve. 
Since,  by  the  definition,  the  effective  value  of  the  equivalent  sine 
wave  is  the  same  as  that  of  the  distorted  wave,  it  follows  that 
this  distorted  wave  of  exciting  current  shares  with  the  sine  wave 
the  feature,  that  the  maximum  value  and  the  effective  value 
have  the  ratio  of  -\/2  -f-  1.  Hence,  below  saturation,  the  maxi- 
mum value  of  the  distorted  curve  can  be  calculated  from  the 
effective  value — which  is  given  by  the  reading  of  an  electro- 


EFFECTIVE  RESISTANCE  AND  REACTANCE    121 


dynamometer — by  using  the  same  ratio  that  applies  to  a  true 
sine  wave,  and  the  magnetic  characteristic  can  thus  be  deter- 
mined by  means  of  alternating  currents,  with  sufficient  exact- 
ness, by  the  electrodynamometer  method,  in  the  range  below 
saturation,  that  is,  by  alternating-current  voltmeter  and  ammeter. 


f   i 
i 


(^1,000  2,000^,000  4,000  5,000  6,000  7,000  8,000   9,000  10,000  11,000  12,000 13,000 14,000 15,000  16,000  17,000. 

FIG.  84. — Magnetization  and  hysteresis  curve. 

96.  In  Fig.  84  is  shown  the  true  magnetic  characteristic  of 
a  sample  of  average  sheet  iron,  as  found  by  the  method  of  slow 
reversals  with  the  magnetometer;  for  comparison  there  is  shown 
in  dotted  lines  the  same  characteristic,  as  determined  with 
alternating  currents  by  the  electrodynamometer,  with  ampere- 


122         ALTERNATING-CURRENT  PHENOMENA 

turns  per  centimeter  as  ordinates  and  magnetic  inductions  as 
abscissas.  As  represented,  the  two  curves  practically  coincide 
up  to  a  value  of  B  =  13,000;  that  is,  up  to  fairly  high  inductions. 
For  higher  saturations,  the  curves  rapidly  diverge,  and  the  elec- 
trodynamometer  curve  shows  comparatively  small  magnetizing 
forces  producing  apparently  very  high  magnetizations. 

The  same  Fig.  84  gives  the  curve  of  hysteretic  loss,  in  ergs 
per  cubic  centimeter  and  cycle,  as  ordinates,  and  magnetic 
inductions  as  abscissas. 

The  electrodynamometer  method  of  determining  the  magnetic 
characteristic  is  preferable  for  use  with  alternating-current 
apparatus,  since  it  is  not  affected  by  the  phenomenon  of  mag- 
netic "creeping,"  which,  especially  at  low  densities,  may  in  the 
magnetometer  tests  bring  the  magnetism  very  much  higher,  or 
the  magnetizing  force  lower,  than  found  in  practice  in  alter- 
nating-current apparatus. 

So  far  as  current  strength  and  power  consumption  are  con- 
cerned, the  distorted  wave  can  be  replaced  by  the  equivalent 
sine  wave  and  the  higher  harmonics  neglected. 

All  the  measurements  of  alternating  currents,  with  the  single 
exception  of  instantaneous  readings,  yield  the  equivalent  sine 
wave  only,  since  all  measuring  instruments  give  either  the  mean 
square  of  the  current  wave  or  the  mean  product  of  instantaneous 
values  of  current  and  e.m.f.,  which,  by  definition,  are  the  same 
in  the  equivalent  sine  wave  as  in  the  distorted  wave. 

Hence,  in  most  practical  applications  it  is  permissible  to 
neglect  the  higher  harmonics  altogether,  and  replace  the  dis- 
torted wave  by  its  equivalent  sine  wave,  keeping  in  mind, 
however,  the  existence  of  a  higher  harmonic  as  a  possible  dis- 
turbing factor  which  may  become  noticeable  in  those  cases  where 
the  frequency  of  the  higher  harmonic  is  near  the  frequency  of 
resonance  of  the  circuit,  that  is,  in  circuits  containing  conden- 
sive  as  well  as  inductive  reactance,  or  in  those  circuits  in  which 
the  higher  harmonic  of  currrent  is  suppressed,  and  thereby  the 
voltage  is  distorted,  as  discussed  in  Chapter  XXV. 

97.  The  equivalent  sine  wave  of  exciting  current  leads  the 
sine  wave  of  magnetism  by  an  angle  QJ,  which  is  called  the  angle 
of  hysteretic  advance  of  phase.  Hence  the  current  lags  behind 
the  e.m.f.  by  the  time  angle  (90°  —  a),  and  the  power  is,  therefore, 

P  =  IE  cos  (90°  -  a)  =  IE  sin  a. 


EFFECTIVE  RESISTANCE  AND  REACTANCE    123 

Thus  the  exciting  current,  /,  consists  of  a  power  component, 
/  sin  a,  called  the  hysteretic  or  magnetic  power  current,  and 
a  wattless  component,  /  cos  a,  which  is  called  the  magnetizing 
current.  Or,  conversely,  the  e.m.f.  consists  of  a  power  compo- 
nent, E  sin  CL,  the  hysteretic  power  component,  and  a  wattless 
component,  E  cos  a,  the  e.m.f.  consumed  by  self-induction. 

Denoting  the  absolute  value  of  the  impedance  of  the  circuit, 

E 

Y>  by  z — where  z  is  determined  by  the  magnetic  characteristic 

of  the  iron  and  the  shape  of  the  magnetic  and  electric  circuits 
— the  impedance  is  represented,  in  phase  and  intensity,  by  the 
symbolic  expression, 

Z  =  r  +  jx  —  z  sin  a  +  jz  cos  a. ; 
and  the  admittance  by, 

1  .1 

Y  =  g  —  fo  =  -  sm  o:  —  j-  cos  a  =  y  sin  a  —  jy  cos  a. 

The  quantities  z,  r,  x,  and  y,  g,  b  are,  however,  not  constants 
as  in  the  case  of  the  circuit  without  iron,  but  depend  upon  the 
intensity  of  magnetization,  B — that  is,  upon  the  e.m.f.  This 
dependence  complicates  the  investigation  of  circuits  containing 
iron. 

In  a  circuit  entirely  inclosed  by  iron,  a  is  quite  considerable, 
ranging  from  30°  to  50°  for  values  below  saturation.  Hence, 
even  with  negligible  true  ohmic  resistance,  no  great  lag  can  be 
produced  in  ironclad  alternating-current  circuits. 

98.  The  loss  of  energy  by  hysteresis  due  to  molecular  magnetic 
friction  is,  with  sufficient  exactness,  proportional  to  the  1.6th 
power  of  magnetic  induction,  B.  Hence  it  can  be  expressed 
by  the  formula: 

W  H  =  r]Bl-« 

where — 

WH—  loss  of  energy  per  cycle,  in  ergs  or  (c.g.s.)  units  (=  10~7 

joules)  per  cubic  centimeter, 

B  =  maximum  magnetic  induction,  in  lines  of  force  per  sq.  cm., 
and  17  =  the  coefficient  of  hysteresis. 

This  I  found  to  vary  in  iron  from  0.001  to  0.0055.  As  a  safe 
mean,  0.0033  can  be  accepted  for  common  annealed  sheet  iron 
or  sheet  steel,  0.002  for  silicon  steel  and  0.0010  to  0.0015  for 
specially  selected  low  hysteresis  steel.  In  gray  cast  iron,  r/  averages 


124         ALTERNATING-CURRENT  PHENOMENA 

0.013;  it  varies  from  0.0032  to  0.028  in  cast  steel,  according  to 
the  chemical  or  physical  constitution;  and  reaches  values  as  high 
as  0.08  in  hardened  steel  (tungsten  and  manganese  steel).  Soft 
nickel  and  cobalt  have  about  the  same  coefficient  of  hysteresis 
as  gray  cast  iron;  in  magnetite  I  found  17  =  0.023. 

In  the  curves  of  Figs.  79  to  84,  rj  =  0.0033. 

At  the  frequency,  /,  the  loss  of  power  in  the  volume,  V,  is,  by 
this  formula, 

p  =  77/751-6  10~7  watts 


jj      10-7  watts, 

where  A  is  the  cross-section  of  the  total  magnetic  flux,  <£. 

The  maximum  magnetic  flux,   <£,  depends  upon  the  counter 
e.m.f.  of  self-induction, 

E  =  \/27rfn3>  10~8, 
tfW 
2wfn' 

where  n  =  number  of  turns  of  the  electric  circuit,  /  =  frequency. 
Substituting  this  in  the  value  of  the  power,  P,  and  canceling, 
we  get, 

E1-6         V  IP5-8  E1'6    F103  t 

™    ~  "n  JUG"  2°-8  irl-GA  1-6n1-6  ~~  ^^ToTe"  ^i.e^i.e* 
or 

MM-6      ,         „  71058  7103 

P  =      .Qg   ,  where  K  =  ^QO.S    1.6.41.6   i.e  =  58??  .16   K6; 

y 
or,  substituting  rj  =  0.0033,  we  have  K  =  191.4^ L61>6; 

or,  substituting  V  =  Al,  where  I  =  length  of  magnetic  circuit, 


58,,110'  _ 

~ 


and 

P  =  — 


/O.G^O.Gyjl.G 

In  Figs.  85,  86,  and  87  is  shown  a  curve  of  hysteretic  loss, 
with  the  loss  of  power  as  ordinates,  and 
in  curve  85,  with  the  e.m.f.,  E}  as  abscissas, 

for  Z  =  6,  A  =  20,  /  =  100,  and  n  =  100; 
in  curve  86,  with  the  number  of  turns  as  abscissas,  for 
I  =  6,  A  =  20,  /  =  100,  and  E  =  100; 


EFFECTIVE  RESISTANCE  AND  REACTANCE    125 


J0U 

170 
160 
1.70 
110 
1.10 
120 
110 
100 
00 
80 
70 

Rl 

:LA 

rioc 

<BE 

TWE 

EN 

EA 

NO 

3 

j 

OR 

1  = 

5,A 

=  20 

,/  = 

-1O 

D.  M 

=  1 

00 

/ 

X 

/ 

cc 

x 

0 

X 

x 

i 

x] 

X 

,50 

40 

x 

<^ 

x 

gX 

,/ 

^ 

^ 

10 
n 

x^ 

^ 

• 

,.  —  • 

,  —  " 

E.N 

l.F. 

FIG.  85. — Hysteresis  loss  as  function  of  E.M.F. 


180 
170 
160 


RELATIO.N    BETWEEN  n  AND  P 
FOR  1  =  6.  A  =  20,  /=100.E  =  100, 


50 


100 


150  200          250  300 

w  =  NUMBE.R  OF  TURNS 

FIG.  86. 


350 


400 


126 


ALTERNATING-CURRENT  PHENOMENA 


in  curve  87,  with  the  frequency,/,  or  the  cross-section,  A,  as 
abscissas,  for  I  =  6,  n  =  100,  and  E  =  100. 

As  shown,  the  hysteretic  loss  is  proportional  to  the  1.6th  power 
of  the  e.m.f.,  inversely  proportional  to  the  1.6th  power  of  the 
number  of  turns,  and  inversely  proportional  to  the  0.6th  power 
of  the  frequency  and  of  the  cross-section. 


RELATION    BETWEEN  N  AND  P 
FOR  A  =  20,  Z=6,n-iOO.E=100 


200 
=  FREQUENCY 

FIG.  87. 


400 


99.  If  g  =  effective  conductance,  the  power  component  of  a 
current  is  /  =  Eg,  and  the  power  consumed  in  a  conductance,  g, 
is  p  =  IE  =  E2g. 

Since,  however, 

#1.6  #1.6 

P  =  K   f06  ,  we  have  K 


it  is: 


ytw 


I 


From  this  we  have  the  following  deduction: 

The  effective  conductance  due  to  magnetic  hysteresis  is  propor- 
tional to  the  coefficient  of  hysteresis,  77,  and  to  the  length  of  the  mag- 
netic circuit,  I,  and  inversely  proportional  to  the  OAth  power  of  the 
e.m.f.,  to  the  0.6th  power  of  the  frequency,  f,  and  of  the  cross-section 


EFFECTIVE  RESISTANCE  AND  REACTANCE    127 

of  the  magnetic  circuit,  A,  and  to  the  1.6th  power  of  the  number  of 
turns,  n. 

Hence,  the  effective  hysteretic  conductance  increases  with 
decreasing  e.m.f.,  and  decreases  with  increasing  e.m.f.;  it  varies, 
however,  much  slower  than  the  e.m.f.,  so  that,  if  the  hysteretic 
conductance  represents  only  a  part  of  the  total  power  consump- 
tion, it  can,  within  a  limited  range  of  variation — as,  for  instance, 
in  constant-potential  transformers — be  assumed  as  constant 
without  serious  error. 


218 
<> 

14 
12 

10 


RELATION  BETWEEN     3     AND    E 
FORZ  =  6,/=100    A=20,w=100 


50 


100 


150    200  E  250 

FIG.  88. 


300 


350 


400 


In  Figs.  88,  89,  and  90,  the  hysteretic  conductance,  g,  is 
plotted,  for  I  =  6,  E  =  100,  /  =  100,  A  =  20  and  n  =  100, 
respectively,  with  the  conductance,  g,  as  ordinates,  and  with 

E  as  abscissas  in  Curve  88. 
/  as  abscissas  in  Curve  89. 
n  as  abscissas  in  Curve  90. 

As  shown,  a  variation  in  the  e.m.f.  of  50  per  cent,  causes  a 
variation  in  g  of  only  14  per  cent.,  while  a  variation  in  /  or  A  by 
50  per  cent,  causes  a  variation  in  g  of  21  per  cent. 

If   (R  =  magnetic    reluctance    of    a   circuit,   FA  =  maximum 


128 


ALTERNATING-CURRENT  PHENOMENA 


RELATION  BETWEEN     0    AND     N 
FOR  J=6,E=100,  A  =  20,  n  =  100 


400 


FIG.  89. 


90 


75 
70 
65 
60 
55 
50 
045 

I" 

35 

30 

25 

20 

15 

10 

6 

0 


RELATION  BETWEEN   n  AND  a 
FOR  f=6.E-100,/=100,  A=20 


100 


150  200  250  300 

W=NUMBER  OF  TURNS 

FIG.  90. 


400 


EFFECTIVE  RESISTANCE  AND  REACTANCE    129 

m.m.f.,  /  =  effective  curren.t,  since  J\/2  =  maximum  current, 
the  magnetic  flux, 

= 
= 


(R  (R 

Substituting  this  in  the  equation  of  the  counter  e.m.f.  of  self- 
induction, 

E  =  V27r/n3>l(T8, 
we  have 

E  " 


hence,  the  absolute  admittance  of  the  circuit  is 

/       (RIO8 


where 


108 

a  =  ~  —  2>  a  constant. 


Therefore,  the  absolute  admittance,  y,  of  a  circuit  of  negligible 
resistance  is  proportional  to  the  magnetic  reluctance,  (R,  and  in- 
versely proportional  to  the  frequency,  f,  and  to  the  square  of  the 
number  of  turns,  n. 

100.  In  a  circuit  containing  iron,  the  reluctance,  (R,  varies  with 
the  magnetization;  that  is,  with  the  e.m.f.  Hence  the  admittance 
of  such  a  circuit  is  not  a  constant,  but  is  also  variable. 

In  an  ironclad  electric  circuit — that  is,  a  circuit  whose  mag- 
netic field  exists  entirely  within  iron,  such  as  the  magnetic  cir- 
cuit of  a  well-designed  alternating-current  transformer — (R  is 
the  reluctance  of  the  iron  circuit.  Hence,  if  /*  =  permeability 
since 

-  «-%-• 

and 

FA  =  IF  =  ^.IH  =  m.m.f., 

<£  =  A(B  =  nAH  =  magnetic  flux, 
and 

10 1  . 

(R  =  T     -r' 


substituting  this  value  in  the  equation  of  the  admittance, 

(RIO8 


130         ALTERNATING-CURRENT  PHENOMENA 
we  have 

no9        c_ 

y-STrWuAf-fn' 
where 

10»         127  1  105 

" 


n*A 

Therefore,  in  an  ironclad  circuit,  the  absolute  admittance,  y,  is 
inversely  proportional  to  the  frequency,  f,  to  the  permeability,  p,  to 
the  cross-section,  A,  and  to  the  square  of  the  number  of  turns,  n\ 
and  directly  proportional  to  the  length  of  the  magnetic  circuit}  I. 

The  conductance  is 

K     . 

y  ~  fo^E0-*' 

and  the  admittance, 


hence,  the  angle  of  hysteretic  advance  is 

g 

sln  a  =      = 


or,  substituting  for  A  and  c  (§119), 

r,l  IP5-8 


Q  a  - 


A°-6n1-6      1  109 
-4n°-4A  °-47r°-4  22-2 


or,  substituting 

E  =  2°-57r/nA(BlO~8, 
we  have 

4  M 
sm  a  =  ^> 

which  is  independent  of  frequency,  number  of  turns,  and  shape 
and  size  of  the  magnetic  and  electric  circuit. 

Therefore,  in  an  ironclad  inductance,  the  angle  of  hysteretic  ad- 
vance, a,  depends  upon  the  magnetic  constants,  permeability  and 
coefficient  of  hysteresis,  and  upon  the  maximum  magnetic  induction, 
but  is  entirely  independent  of  the  frequency,  of  the  shape  and  other 
conditions  of  the  magnetic  and  electric  circuit,'  and,  therefore,  all 
ironclad  magnetic  circuits  constructed  of  the  same  quality  of  iron 
and  using  the  same  magnetic  density,  give  the  same  angle  of  hys- 
teretic advance,  and  the  same  power  factor  of  their  electric  energizing 
circuit. 


EFFECTIVE  RESISTANCE  AND  REACTANCE    131 

The  angle  of  hysteretic  advance,  a,  in  a  closed  circuit  trans- 
former and  the  no-load  power  factor,  depend  upon  the  quality  of  the 
iron,  and  upon  the  magnetic  density  only. 

The  sine  of  the  angle  of  hysteretic  advance  equals  4  times  the 
product  of  the  permeability  and  coefficient  of  hysteresis,  divided  by 
the  0  .  4th  power  of  the  magnetic  density. 

101.  If  the  magnetic  circuit  is  not  entirely  ironclad,  and  the 
magnetic  structure  contains  air-gaps,  the  total  reluctance  is 
the  sum  of  the  iron  reluctance  and  of  the  air  reluctance,  or 

(R  =  (Ri  +  (Ra; 

hence  the  admittance  is 

%  * 

y  =  Vg2  +  &2  =    (<JU  +  (R«). 


Therefore,  in  a  circuit  containing  iron,  the  admittance  is  the 

/•»  /Q   . 

~pj 


/•»  /Q   . 

sum  of  the  admittance  due  to  the  iron  part  of  the  circuit,  ?/»  =  ~ 


and  of  the  admittance  due  to  the  air  part  of  the  circuit,  y 

if  the  iron  and  the  air  are  in  series  in  the  magnetic  circuit. 

The  conductance,  g,  represents  the  loss  of  power  in  the  iron, 
and,  since  air  has  no  magnetic  hysteresis,  is  not  changed  by  the 
introduction  of  an  air-gap.  Hence  the  angle  of  hysteretic 
advance  of  phase  is 


Vi   (Rt  +   <Ra 

and  a  maximum,  —  ,  for  the  ironclad  circuit,  but  decreases  with 

Vi 
increasing  width  of  the  air-gap.     The  introduction  of  the  air- 

gap  of  reluctance,  (Ra,  decreases  sin  a  in  the  ratio, 

(Hi 


In  the  range  of  practical  application,  from  B  =  2,000  to 
B  =  14,000,  the  permeability  of  iron  usually  exceeds  1,000,  while 
sin  a  in  an  ironclad  circuit  varies  in  this  range  from  0.51  to  0.69. 
In  air,  /*  =  1. 

If,  consequently,  1  per  cent,  of  the  length  of  the  iron  consists 
of  an  air-gap,  the  total  reluctance  only  varies  by  a  few  per  cent., 
that  is,  remains  practically  constant;  while  the  angle  of  hysteretic 
advance  varies  from  sin  a  =  0.035  to  sin  a  =  0.064.  Thus  g 
is  negligible  compared  with  6;  and  b  is  practically  equal  to  y. 


132         ALTERNATING-CURRENT  PHENOMENA 

Therefore,  in  an  electric  circuit  containing  iron,  but  forming 
an  open  magnetic  circuit  whose  air-gap  is  not  less  than  34  oo  the 
length  of  the  iron,  the  susceptance  is  practically  constant  and 
equal  to  the  admittance,  so  long  as  saturation  is  not  yet  ap- 
proached, or, 

,  <Ra  / 

=  r  or:  *  =  ^ 

The  angle  of  hysteretic  advance  is  small,  and  the  hysteretic  con- 
ductance is 

K 


The  current  wave  is  practically  a  sine  wave. 

As  an  example,  in  Fig.  83,  Curve  II,  the  current  curve  of  a 
circuit  is  shown,  containing  an  air-gap  of  only  34  oo  of  the  length 
of  the  iron,  giving  a  current  wave  much  resembling  the  sine 
shape,  with  an  hysteretic  advance  of  9°. 

102.  To  determine  the  electric  constants  of  a  circuit  con- 
taining iron,  we  shall  proceed  in  the  following  way: 

Let 

E  =  counter  e.m.f.  of  self-induction 

then  from  the  equation, 

E  =  V2  7rn/$10-8, 

where  /  =  frequency,  n  =  number  of  turns, 

we  get  the  magnetism,  <l>,  and  by  means  of  the  magnetic  cross- 

& 
section,  A,  the  maximum  magnetic  induction:  B  =  -T- 

From  B,  we  get,  by  means  of  the  magnetic  characteristic  of 
the  iron,  the  magnetizing  force,  =  /  ampere-turns  per  centimeter 
length  where 


if  H  =  magnetizing  force  in  c.g.s.  units. 

Hence, 

if  h  =  length  of  iron  circuit,  F»  =  lif  =  ampere-turns  required 
in  the  iron; 

if  la  =  length  of  air  circuit,  Fa  =  ~r~~  —  ampere-turns  required 
in  the  air: 


EFFECTIVE  RESISTANCE  AND  REACTANCE    133 

hence,  F  =  F*  +  Fa  =  total  ampere-turns,  maximum  value,  and 

F 

—.=  =  effective  value.     The  exciting  current  is 

V2 

w\/2 
and  the  absolute  admittance, 


y  =  =    - 

If  F»  is  not  negligible  as  compared  with  Fa,  this  admittance,  y, 
is  variable  with  the  e.m.f.,  E. 

If  V  =  volume  of  iron,  77  =  coefficient  of  hysteresis, 
the   loss   of   power   by   hysteresis   due   to   molecular   magnetic 
friction  is 

P  m  77/FS1-6; 

p 
hence  the  hysteretic  conductance  is  g  =  ™,    and    variable    with 

the  e.m.f.,  E. 

The  angle  of  hysteretic  advance  is 

sin  a  =  -; 

y 

the  susceptance. 


the  effective  resistance, 
and  the  reactance, 


X    =     — 2 

y2 


103.  As  conclusions,  we  derive  from  this  chapter  the  following: 

1.  In  an  alternating-current  circuit  surrounded  by  iron,  the 
current  produced  by  a  sine  wave  of  e.m.f.  is  not  a  true  sine  wave, 
but  is  distorted  by  Hysteresis,  and  inversely,  a  sine  wave  of 
current  requires  waves  of  magnetism  and  e.m.f.  differing  from 
sine  shape. 

2.  This  distortion  is  excessive  only  with  a  closed  magnetic 
circuit  transferring  no  energy  into  a  secondary  circuit  by  mutual 
inductance. 

3.  The  distorted  wave  of  current  can  be  replaced  by  the  equiva- 
lent sine  wave — that  is,  a  sine  wave  of  equal  effective  intensity 
and   equal   power — and  the  superposed  higher  harmonic,  con- 


134         ALTERNATING-CURRENT  PHENOMENA 

sisting  mainly  of  a  term  of  triple  frequency,  may  be  neglected 
except  in  resonating  circuits. 

4.  Below  saturation,  the  distorted  curve  of  current  and  its 
equivalent  sine  wave  have  approximately  the  same  maximum 
value. 

5.  The  angle  of  hysteretic  advance — that  is,  the  phase  dif- 
ference between  the  magnetic  flux  and  equivalent  sine  wave  of 
m.m.f. — is  a  maximum  for  the   closed   magnetic   circuit,    and 
depends  there  only  upon  the  magnetic  constants  of  the  iron,  upon 
the  permeability,  ju,  the  coefficient  of  hysteresis,  r?,  and  the  maxi- 
mum magnetic  induction,  as  shown  in  the  equation, 

4  AIT; 
sin  a  =  g^j- 

6.  The  effect  of  hysteresis  can  be  represented  by  an  admittance 
Y  =  g  —  jb,  or  an  impedance,  Z  =  r  +  jx. 

7.  The    hysteretic    admittance,    or    impedance,    varies    with 
the  magnetic  induction;  that  is,  with  the  e.m.f.,  etc. 

8.  The    hysteretic    conductance,    g,    is    proportional    to    the 
coefficient  of  hysteresis,  rj,  and  to  the  length  of  the  magnetic 
circuit,  I,  inversely  proportional  to  the  0.4th  power  of  the  e.m.f. 
E,  to  the  0.6th  power  of  frequency, /,  and  of  the  cross-section  of 
the  magnetic  circuit,  A,  and  to  the  1.6th  power  of  the  number  of 
turns  of  the  electric  circuit,  n,  as  expressed  in  the  equation, 

58  7,1 103 

9    "    ^0.4/0.6^0.6^1.6* 

9.  The  absolute  value  of  hysteretic  admittance, 


y  = 

is  proportional  to  the  magnetic  reluctance:  (R  =  (R»  +  (Ra,  and 
inversely  proportional  to  the  frequency,  /,  and  to  the  square  of 
the  number  of  turns,  n,  as  expressed  in  the  equation, 

_  ((Rj  +  (Ra)  108 
y  =          2irfn* 

10.  In  an  ironclad  circuit,  the"  absolute  value  of  admittance 
is  proportional  to  the  length  of  the  magnetic  circuit,  and  inversely 
proportional  to  cross-section,  A,  frequency,  /,  permeability,  n 
and  square  of  the  number  of  turns,  n,  or 

127  Z 105 
Vi 


11.  In  an  open  magnetic  circuit,  the  conductance,  g,  is  the 
same  as  in  a  closed  magnetic  circuit  of  the  same  iron  part. 


EFFECTIVE  RESISTANCE  AND  REACTANCE    135 

12.  In  an  open  magnetic  circuit,  the  admittance,  y,  is  prac- 
tically constant,  if  the  length  of  the  air-gap  is  at  least  ^oo  of  the 
length  of  the  magnetic  circuit,  and  saturation  be  not  approached. 

13.  In  a  closed  magnetic  circuit,  conductance,  susceptance, 
and  admittance  can  be  assumed  as  constant  through  a  limited 
range  only. 

14.  From  the  shape  and  the  dimensions  of  the  circuits,  and 
the  magnetic  constants  of  the  iron,  all  the  electric  constants,  g,  6, 
y,r,x,  z,  can  be  calculated. 

104.  The  preceding  applies  to  the  alternating  magnetic  circuit, 
that  is,  circuit  in  which  the  magnetic  induction  varies  between 
equal  but  opposite  limits:  BI  =  +  BQ  and  B2  =  —  BQ. 

In  a  pulsating  magnetic  circuit,  in  which  the  induction  B  varies 
between  two  values  BI  and  B2}  which  are  not  equal  numerically, 
and  which  may  be  of  the  same  sign  or  of  opposite  sign,  that  is 
in  which  the  hysteresis  cycle  is  unsymmetrical,  the  law  of  the 
1.6th  power  still  applies,  and  the  loss  of  energy  per  cycle  is  pro- 
portional to  the  1.6th  power  of  the  amplitude  of  the  magnetic 
variation  : 


but  the  hysteresis  coefficient  t]  is  not  the  same  as  for  alternating 

magnetic  circuits,   but  increases  with  increasing  average  value 
P     i    p 

—  '-^  -  -  of  the  magnetic  induction. 

Zi 

Such  unsymmetrical  magnetic  cycles  occur  in  some  types  of 
induction  alternators,1  in  which  the  magnetic  induction  does  not 
reverse,  but  pulsates  between  a  high  and  a  low  value  in  the 
same  direction. 

Unsymmetrical  magnetic  cycles  occasionally  occur  —  and  give 
trouble  —  in  transformers  by  the  entrance  of  a  stray  direct  current 
(railway  return)  over  the  ground  connection,  or  when  an  unsuit- 
able transformer  connection  is  used  on  a  synchronous  converter 
feeding  a  three-wire  system. 

Very  unsymmetrical  cycles  may  give  very  much  higher  losses 
than  symmetrical  cycles  of  the  same  amplitude. 

For  more  complete  discussion  of  unsymmetrical  cycles  see 
"Theory  and  Calculation  of  Electric  Circuits." 

1  See  "Theory  and  Calculation  of  Electric  Apparatus." 


CHAPTER  XIII 
FOUCAULT  OR  EDDY  CURRENTS 

105.  While  magnetic  hysteresis  due  to  molecular  friction  is  a 
magnetic  phenomenon,  eddy  currents  are  rather  an  electrical 
phenomenon.  When  iron  passes  through  a  magnetic  field,  a 
loss  of  energy  is  caused  by  hysteresis,  which  loss,  however, 
does  not  react  magnetically  upon  the  field.  When  cutting  an 
electric  conductor,  the  magnetic  field  produces  a  current  therein. 
The  m.m.f.  of  this  current  reacts  upon  and  affects  the  magnetic 
field,  more  or  less;  consequently,  an  alternating  magnetic  field 
cannot  penetrate  deeply  into  a  solid  conductor,  but  a  kind  of 
screening  effect  is  produced,  which  makes  solid  masses  of  iron 
unsuitable  for  alternating  fields,  and  necessitates  the  use  of 
laminated  iron  or  iron  wire  as  the  carrier  of  magnetic  flux. 

Eddy  currents  are  true  electric  currents,  though  existing  in 
minute  circuits;  and  they  follow  all  the  laws  of  electric  circuits. 

Their  e.m.f.  is  proportional  to  the  intensity  of  magnetization, 
B,  and  to  the  frequency,  /. 

Eddy  currents  are  thus  proportional  to  the  magnetization, 
By  the  frequency,  /,  and  to  the  electric  conductivity,  X,  of  the 
iron;  hence,  can  be  expressed  by 

i  =  b\Bf. 

The  power  consumed  by  eddy  currents  is  proportional  to 
their  square,  and  inversely  proportional  to  the  electric  conduc- 
tivity, and  can  be  expressed  by 

P  = 


or,  since  Bf  is  proportional  to  the  generated  e.m.f.,  E,  in  the 
equation 

E  =  VZirAufBlQ-*, 

it  follows  that,  The  loss  of  power  by  eddy  currents  is  proportional 
to  the  square  of  the  e.m.f.,  and  proportional  to  the  electric  con- 
ductivity of  the  iron;  or, 

P  =  aEz\. 
136 


FOUCAULT  OR  EDDY  CURRENTS  137 

Hence,  that  component  of  the  effective  conductance  which 
is  due  to  eddy  currents  is 

P 

®  ==  W2  =     ' 

that  is,  The  equivalent  conductance  due  to  eddy  currents  in  the 
iron  is  a  constant  of  the  magnetic  circuit;  it  is  independent  of 
e.m.f.,  frequency,  etc.,  but  proportional  to  the  electric  conductivity 
of  the  iron,  X. 

Eddy  currents,  like  magnetic  hysteresis,  cause  an  advance  of 
phase  of  the  current  by  an  angle  of  advance,  /? ;  but  unlike 
hysteresis,  eddy  currents  in  general  do  not  distort  the  current 
wave. 

The  angle  of  advance  of  phase  due  to  eddy  currents  is 

sin  p  =  -  , 

where  y  =  absolute  admittance  of  the  circuit,  g  =  eddy  current 
conductance. 

While  the  equivalent  conductance,  g,  due  to  eddy  currents, 
is  a  constant  of  the  circuit,  and  independent  of  e.m.f.,  frequency, 
etc.,  the  loss  of  power  by  eddy  currents  is  proportional  to  the 
square  of  the  e.m.f.  of  self-induction,  and  therefore  proportional 
to  the  square  of  the  frequency  and  to  the  square  of  the 
magnetization. 

Only  the  power  component,  gE,  of  eddy  currents,  is  of  interest, 
since  the  wattless  component  is  identical  with  the  wattless  com- 
ponent of  hysteresis,  discussed  in  the  preceding  chapter. 

106.  To  calculate  the  loss  of  power  by  eddy  currents, 

Let  V  =  volume  of  iron ; 

B  =  maximum  magnetic 'induction; 
/  =  frequency; 

X  =  electric  conductivity  of  iron; 
c  =  coefficient  of  eddy  currents. 

The  loss  of  energy  per  cubic  centimeter,  in  ergs  per  cycle,  is 

w  =  eX/£2; 
hence,  the  total  loss  of  power  by  eddy  currents  is 

P  =  eXF/2B2 10-7  watts, 
and  the  equivalent  conductance  due  to  eddy  currents  is 

P_         IQeXZ        0.507  €\l 
g  ~  E2  ~  2ir2An2  ~      An2 


138         ALTERNATING-CURRENT  PHENOMENA 


where 

I  =  length  of  magnetic  circuit, 
A  =  section  of  magnetic  circuit, 
n  =  number  of  turns  of  electric  circuit. 

The  coefficient  of  eddy  currents,  e,  depends  merely  upon  the 
shape  of  the  constituent  parts  of  the  magnetic  circuit;  that  is, 
whether  of  iron  plates  or  wire,  and  the  thickness 
of  plates  or  the  diameter  of  wire,  etc. 

The  two  most  important  cases  are: 

(a)  Laminated  iron. 
(6)  Iron  wire. 

107.  (a)  Laminated  Iron. 
Let,  in  Fig.  91, 

d  =  thickness  of  the  iron  plates; 

B  =  maximum  magnetic  induction; 

/  =  frequency; 

X  =  electric  conductivity  of  the  iron. 

Then,  if  u  is  the  distance  of  a  zone,  du,  from 
the  center  of  the  sheet,  the  conductance  of  a 
zone  of  thickness,  du,  and  of  one  centimeter 
length  and  width  is  \du;  and  the  magnetic  flux 
cut  by  this  zone  is  Bu.  Hence,  the  e.m.f. 
induced  in  this  zone  is 

5E  =  \/2  irfBu,  in  c.g.s.  units.  FIG.  91. 

This  e.m.f.  produces  the  current,  dl  =  dE  \du  =  \/2  irfB  udu, 
in  c.g.s.  units,  provided  the  thickness  of  the  plate  is  negligible  as 
compared  with  the  length,  in  order  that  the  current  may  be 
assumed  as  parallel  to  the  sheet,  and  in  opposite  directions  on 
opposite  sides  of  the  sheet. 

The  power  consumed  by  the  current  in  this  zone,  du,  is 

dP  =  dEdl  =  27T2/2£2Xw2dtt, 

in  c.g.s.  units  or  ergs  per  second,  and,  consequently,  the  total 
power  consumed  in  one  square  centimeter  of  the  sheet  of  thick- 
ness, d,  is 


d 

i 

j 

i 

! 

! 

i 

i 

"i 

=  27r2/2£2; 


. 
,  in  c.g.s.  units; 


FOUCAULT  OR  EDDY  CURRENTS  139 

the  power  consumed  per  cubic  centimeter  of  iron  is,  therefore, 

5P       Tr2f2B2\d~    . 
p  =  —T  =  --  ~  --  ,  in  c.g.s.  units  or  erg-seconds, 

and  the  energy  consumed  per  cycle  and  per  cubic  centimeter  of 
iron  is 


p 
w  =  j  --       —  g  --  ergs. 

The  coefficient  of  eddy  currents  for  laminated  iron  is,  therefore, 

TT2d2 

e  =  -g-  =  1.645  d2, 

where  X  is  expressed  in  c.g.s.  units.     Hence,  if  X  is  expressed  in 
practical  units  or  10~9  c.g.s.  units, 


Substituting  for  the  conductivity  of  sheet  iron  the  approxi- 
mate value. 

X  =  105,1 
we  get  as  the  coefficient  of  eddy  currents  for  laminated  iron, 

e  =  ^  d2  10~9  =  1.645  d2  10~9; 
loss  of  energy  per  cubic  centimeter  and  cycle, 

W  =  e\fB2  =  ^  d2\fB2  10~9  =  1.645  d*\fB2  10~9  ergs 

=  1.  645  d2fB*lQ-*  ergs; 
or,  W  =  eX/£210-7  =  1.645  d2/£210-n  joules. 
The  loss  of  power  per  cubic  centimeter  at  frequency,  /,  is 

p  =  fW  =  eXfB^O-7  =  1.645  d2/2£210-"  watts; 
the  total  loss  of  power  in  volume,  V,  is 

P  =  Vp  =  1.645  Vd2f2B210-n  watts. 
As  an  example, 

d  =  1  mm.  =  0.1  cm.;/  =  100;  B  =  5,000;  V  =  1,000  c.c.; 
e  =  1,645  X  10-"; 
W  =  4,1  10  ergs 

=  0.000411  joules; 
p  =  0.0411  watts; 
P  =  41.4  watts. 

1  In  some  of  the  modern  silicon  steels  used  for  transformer  iron,  X  reaches 
values  as  low  as  2  X  104,  and  even  lower;  and  the  eddy  current  losses  are 
reduced  in  the  same  proportion  (1915). 


140         ALTERNATING-CURRENT  PHENOMENA 

108.  (6)  Iron  Wire. 

Let,  in  Fig.  92,  d  =  diameter  of  a  piece  of  iron  wire;  then  if 
u  is  the  radius  of  a  circular  zone  of  thickness,  du,  and  one  cen- 
timeter in  length,  the  conductance  of  this  zone  is  ~ — ,  and  the 
magnetic  flux  inclosed  by  the  zone  is  Bu*ir. 


FIG.  92. 


Hence,  the  e.m.f.  generated  in  this  zone  is 

BE  =  \/2ir2fBu2  in  c.g.s.  units, 
and  the  current  produced  thereby  is 


\fBu  du,  in  c.g.s.  units. 

i 

The  power  consumed  in  this  zone  is,  therefore, 

dP  =  8EdI  =  Tr*\f*B2usdu,  in  c.g.s.  units; 

consequently,  the  total  power  consumed  in  one  centimeter  length 
of  wire  is 


=    (  2dW  =  T^XPB2  ( 


,  in  c.g.s.  units. 

Since  the  volume  of  one  centimeter  length  of  wire  is 

(Pr 

T' 

the  power  consumed  in  one  cubic  centimeter  of  iron  is 

p  =  --  =  —  \f*B2d2,  in  c.g.s.  units  or  erg-seconds, 


FOUCAULT  OR  EDDY  CURRENTS  141 

and  the  energy  consumed  per  cycle  and  cubic  centimeter  of  iron  is 

w  =     =      x/B2rf2  ergs* 


Therefore,  the  coefficient  of  eddy  currents  for  iron  wire  is 
e  =  |?<fi  =  0.617  d2; 


=  ^  d2 10~9  -  0.617  d2  10~9. 


or,  if  X  is  expressed  in  practical  units,  or  10~9  c.g.s.  units, 

Substituting 

X  =  105, 

we  get  as  the  coefficient  of  eddy  currents  for  iron  wire, 

e  =  ^  d2  10~9  =  0.617  d2  10~9. 
16 

The  loss  of  energy  per  cubic  centimeter  of  iron,  and  per  cycle, 
becomes 

W  =  eX/£2  =  ~  d2\fB2  10-9  =  0.617  d2\fB2  10~9 

=  0.617  d2fB2  10~4  ergs, 

=  eX/52  10~7  =  0.617  d2fB2  W~n  joules; 

loss  of  power  per  cubic  centimeter  at  frequency,  /, 

p  =  fW  =  t\N2B2  10-7  =  0.617  d2N2B2  10~n  watts; 
total  loss  of  power  in  volume,  V, 

p  =  Vp  =  0.617  Vd^B2  10-11  watts. 
As  an  example, 
d  =  1  mm.,  =  0.1  cm.;  /  =  100;  B2  =  5,000;  V  =1,000  cu.  cm. 

Then, 

e  =  0.617  X  10-11, 
W  =  1,540  ergs  =  0.000154  joules, 
p  =  0.0154  watts, 
P  =  1.54  watts, 

hence  very  much  less  than  in  sheet  iron  of  equal  thickness. 
109.  Comparison  of  sheet  iron  and  iron  wire. 
If 

di  =  thickness  of  lamination  of  sheet  iron,  and 

dz  =  diameter  of  iron  wire, 


142         ALTERNATING-CURRENT  PHENOMENA 
the  eddy  current  coefficient  of  sheet  iron  being 

6!  =  ^  dS  10-9, 
and  the  eddy  current  coefficient  of  iron  wire 


the  loss  of  power  is  equal  in  both  —  other  things  being  equal  —  if 
ci  =  €2;  that  is,  if 

d22  =  %di*,  or  d2  =  1.63  di. 

It  fojlows  that  the  diameter  of  iron  wire  can  be  1.63  times  or, 
roughly,  1%  as  large  as  the  thickness  of  laminated  iron,  to  give 
the  same  loss  of  power  through  eddy  currents,  as  shown  in  Fig. 
93. 


FIG.  93. 

110.  Demagnetizing,  or  screening  effect  of  eddy  currents. 

The  formulas  derived  for  the  coefficient  of  eddy  currents  in 
laminated  iron  and  in  iron  wire  hold  only  when  the  eddy  currents 
are  small  enough  to  neglect  their  magnetizing  force.  Other- 
wise the  phenomenon  becomes  more  complicated;  the  magnetic 
flux  in  the  interior  of  the  lamina,  or  the  wire,  is  not  in  phase  with 
the  flux  at  the  surface,  but  lags  behind  it.  The  magnetic  flux 
at  the  surface  is  due  to  the  impressed  m.m.f.,  while  the  flux  in  the 
interior  is  due  to  the  resultant  of  the  impressed  m.m.f.  and  to  the 
m.m.f.  of  eddy  currents;  since  the  eddy  currents  lag  90  degrees 
behind  the  flux  producing  them,  their  resultant  with  the 
impressed  m.m.f.,  and  therefore  the  magnetism  in  the  interior, 
is  made  lagging.  Thus,  progressing  from  the  surface  toward 
the  interior,  the  magnetic  flux  gradually  lags  more  and  more  in 
phase,  and  at  the  same  time  decreases  in  intensity.  While  the 
complete  analytical  solution  of  this  phenomenon  is  beyond  the 


FOUCAULT  OR  EDDY  CURRENTS  143 

scope  of  this  book,  a  determination  of  the  magnitude  of  this 
demagnetization,  or  screening  effect,  sufficient  to  determine 
whether  it  is  negligible,  or  whether  the  subdivision  of  the  iron 
has  to  be  increased  to  make  it  negligible,  can  be  made  by  calcu- 
lating the  maximum  magnetizing  effect,  which  cannot  be  exceeded 
by  the  eddys. 

Assuming  the  magnetic  density  as  uniform  over  the  whole 
cross-section,  and  therefore  all  the  eddy  currents  in  phase  with 
each  other,  their  total  m.m.f.  represents  the  maximum  possible 
value,  since  by  the  phase  difference  and  the  lesser  magnetic 
density  in  the  center  the  resultant  m.m.f.  is  reduced. 

In  laminated  iron  of  thickness  d,  the  current  in  a  zone  of  thick- 
ness du,  at  distance  u  from  center  of  sheet,  is 

dl  =  \/2  irfB\u  du  units  (c.g.s.) 
=  \/2 TtfB\u  du  10~8  amp.; 

hence  the  total  current  in  the  sheet  is 

ri  ri 

I  =  I     dl  =  \/2  7r/£X  10~8  I     u  du 
Jo  Jo 


Hence,    the    maximum    possible    demagnetizing    ampere-turns, 
acting  upon  the  center  of  the  lamina,  are 

1(T8  =  0.555/£Xd2  1(T8, 

o 

=  0.555  fB\d2  10~8  ampere-turns  per  cm. 

Example:  d  =  0.1  cm.,/  =  100,  B  =  5000,  X  =  105, 
or  /  =  2.775  ampere-turns  per  cm. 

111.  In  iron  wire  of  diameter  d,  the  current  in  a  tubular  zone 
of  du  thickness  and  u  radius  is 


dl  =      -  irfBXu  du  10~8  amp.; 
hence,  the  total  current  is 


8  y  u 


I  =         dl  =  ~TrfB\  10~8         u  dx 

A/2 
=  -T-  7r/£Xd2  10~8  amp. 


144         ALTERNATING-CURRENT  PHENOMENA 

Hence,    the    maximum    possible    demagnetizing    ampere-turns, 
acting  upon  the  center  of  the  wire,  are 


/  =  /£Xd2  10-8  =  0.2775  fB\d*  10~8 

=  0.2775  fB\d2  10~8  ampere-turns  per  cm. 

For  example,  if  d  =  0.1  cm.,/  =  100,  B  =  5000,  X  =  105,  then 
/  =  1.338  ampere-turns  per  cm.;  that  is,  half  as  much  as  in  a 
lamina  of  the  thickness  d. 

For  a  more  complete  investigation  of  the  screening  effect  of 
eddy  currents  in  laminated  iron,  see  Section  III  of  "  Theory  and 
Calculation  of  Transient  Electric  Phenomena  and  Oscillations.  " 

112.  Besides  the  eddy,  or  Foucault,  currents  proper,  which 
exist  as  parasitic  currents  in  the  interior  of  the  iron  lamina  or 
wire,  under  certain  circumstances  eddy  currents  also  exist  in 
larger  orbits  from  lamina  to  lamina  through  the  whole  magnetic 
structure.     Obviously  a  calculation  of  these  eddy  currents  is 
possible  only  in  a  particular  structure.     They  are  mostly  surface 
currents,  due  to  short  circuits  existing  between  the  laminae  at 
the  surface  of  the  magnetic  structure. 

Furthermore,  eddy  currents  are  produced  outside  of  the  mag- 
netic iron  circuit  proper,  by  the  magnetic  stray  field  cutting 
electric  conductors  in  the  neighborhood,  especially  when  drawn 
toward  them  by  iron  masses  behind,  in  electric  conductors 
passing  through  the  iron  of  an  alternating  field,  etc.  All  these 
phenomena  can  be  calculated  only  in  particular  cases,  and  are  of 
less  interest,  since  they  can  and  should  be  avoided. 

The  power  consumed  by  such  large  eddy  currents  frequently 
increases  more  than  proportional  to  the  square  of  the  voltage, 
when  approaching  magnetic  saturation,  by  the  magnetic  stray 
field  reaching  unlaminated  conductors,  and  so,  while  negligible 
at  normal  voltage,  this  power  may  become  large  at  over-normal 
voltage. 

Eddy  Currents  in  Conductor,  and  Unequal  Current  Distribution 

113.  If  the  electric  conductor  has  a  considerable  size,  the 
alternating  magnetic  field,  in  cutting  the  conductor,  may  set 
up  differences  of  potential  between  the  different  parts  thereof, 
thus  giving  rise  to  local  or  eddy  currents  in  the  copper.     This 
phenomenon  can  obviously  be  studied  only  with  reference  to  a 


FOUCAULT  OR  EDDY  CURRENTS  145 

particular  case,  where  the  shape  of  the  conductor  and  the  dis- 
tribution of  the  magnetic  field  are  known. 

Only  in  the  case  where  the  magnetic  field  is  produced  by  the 
current  in  the  conductor  can  a  general  solution  be  given.  The 
alternating  current  in  the  conductor  produces  a  magnetic  field, 
not  only  outside  of  the  conductor,  but  inside  of  it  also;  and  the 
lines  of  magnetic  force  which  close  themselves  inside  of  the  con- 
ductor generate  e.m.fs.  in  their  interior  only.  Thus  the  counter 
e.m.f.  of  self-induction  is  largest  at  the  axis  of  the  conductor,  and 
least  at  its  surface;  consequently,  the  current  density  at  the  sur- 
face will  be  larger  than  at  the  axis,  or,  in  extreme  cases,  the  cur- 
rent may  not  penetrate  at  all  to  the  center,  or  a  reversed  current 
may  exist  there.  Hence  it  follows  that  only  the  exterior  part 
of  the  conductor  may  be  used  for  the  conduction  of  electricity, 
thereby  causing  an  increase  of  the  ohmic  resistance  due  to  unequal 
current  distribution.  * 

The  general  discussion  of  this  problem,  as  applicable  to  the 
distribution  of  alternating  current  in  very  large  conductors, 
as  the  iron  rails  of  the  return  circuit  of  alternating-current  rail- 
ways, is  given  in  Section  III  of  "  Theory  and  Calculation  of  Tran- 
sient Electric  Phenomena  and  Oscillations." 

In  practice,  this  phenomenon  is  observed  mainly  with  very 
high  frequency  currents,  as  lightning  discharges,  wireless  tele- 
graph and  lightning  arrester  circuits;  in  power-distribution  cir- 
cuits it  has  to  be  avoided  by  either  keeping  the  frequency  suffi- 
ciently low  or  having  a  shape  of  conductor  such  that  unequal 
current-distribution  does  not  take  place,  as  by  using  a  tubular  or  a 
flat  conductor,  or  several  conductors  in  parallel. 

114.  It  will,  therefore,  here  be  sufficient  to  determine  the 
largest  size  of  round  conductor,  or  the  highest  frequency,  where 
this  phenomenon  is  still  negligible. 

In  the  interior  of  the  conductor,  the  current  density  is  not 
only  less  than  at  the  surface,  but  the  current  lags  in  phase  be- 
hind the  current  at  the  surface,  due  to  the  increased  effect  of 
self-induction.  This  time-lag  of  the  current  causes  the  magnetic 
fluxes  in  the  conductor  to  be  out  of  phase  with  each  other,  making 
their  resultant  less  than  their  sum,  while  the  lesser  current  density 
in  the  center  reduces  the  total  flux  inside  of  the  conductor.  Thus, 
by  assuming,  as  a  basis  for  calculation,  a  uniform  current  density 
and  no  difference  of  phase  between  the  currents  in  the  different 

layers  of  the  conductor,  the  unequal  distribution  is  found  larger 
10 


146 


ALTERNATING-CURRENT  PHENOMENA 


than  it  is  in  reality.  Hence  this  assumption  brings  us  on  the  safe 
side,  and  at  the  same  time  greatly  simplifies  the  calculation; 
however,  it  is  permissible  only  where  the  current  density  is  still 
fairly  uniform. 

Let  Fig.  94  represent  a  cross-section  of  a  conductor  of  radius, 
R,  and  a  uniform  current  density, 

.         7 


where  7  =  total  current  in  conductor. 

The  magnetic  reluctance  of  a  tubular 
zone  of  unit  length  and  thickness  du, 
•~   of  radius  u,  is 
2  UTT 


(R, 


du 


FIG.  94. 


The  current  inclosed  by  this  zone  is  7M 
=  iu2ir,  and  therefore,  the  m.m.f.  acting 
upon  this  zone  is 
Fu  =  0.4  irlu  =  0.4  7r2m2, 
and  the  magnetic  flux  in  this  zone  is 

d&  =  — -  =  0.27riu  du. 
Hence,  the  total  magnetic  flux  inside  the  conductor  is 

From  this  we  get,  as  the  excess  of  counter  e.m.f.  at  the  axis  of 
the  conductor  over  that  at  the  surface, 

A#  =  V2Trf3>  10~8  =  \/2  irfl  10~9,  per  unit  length,      . 

=  V2^fiR2 10~9; 

and  the  reactivity,  or  specific  reactance  at  the  center  of  the  con- 
ductor, becomes  k 


A     Tjl  ± 

-  -  =  A/2  ?r2/#2  10~9. 


Let  p  =  resistivity,  or  specific  resistance,  of  the  material  of  the 
conductor. 

We  have  then, 


and 


P 
p 


the  ratio  of  current  densities  at  center  and  at  periphery. 


FOUCAULT  OR  EDDY  CURRENTS  147 

For  example,  if,  in  copper,  p  =  1.7  X  10~6,  and  the  percentage 
decrease  of  current  density  at  center  shall  not  exceed  5  per  cent., 

that  is, 

P  •*•  \A2  +  p2  =  0.95  -r-  1, 
we  have 

k  =  0.51  X  10~6; 
hence 

0.51  X  10~6  =  V2ir2/R810~9, 
or 

fR*  =  36.3; 
hence,  when 

/  =       125         100         60  25 

R  =  0.541      0.605      0.781      1.21  cm. 
D  =  2R  =  1.08        1.21        1.56        2.42  cm.' 

Hence,  even  at  a  frequency  of  125  cycles,  the  effect  of  unequal 
current  distribution  is  still  negligible  at  one  centimeter  diameter 
of  the  conductor.  Conductors  of  this  size  are,  however,  excluded 
from  use  at  this  frequency  by  the  external  self-induction,  which 
is  several  times  larger  than  the  resistance.  We  thus  see  that  un- 
equal current  distribution  is  usually  negligible  in  practice. 

The  above  calculation  was  made  under  the  assumption  that 
the  conductor  consists  of  unmagnetic  material.  If  this  is  not 
the  case,  but  the  conductor  of  iron  of  permeability 

M,  then  d3>  =  ^-Ji;  and  thus  ultimately,  k  =  \/2  irzfaRz  10~9,and 
CKw 

-  =  \/2  7T2  —       — .     Thus,  for  instance,  for  iron  wire  at  p  = 
P  P 

10  X  10~6,  p  =  500,  it  is,  permitting  5  per  cent,  difference  be- 
tween center  and  outside  of  wire,  k  =  3.2  X  10~6,  and  fR2  = 
0.46; 
hence,  when 

/  =       125         100         60  25 

fl  =   0.061      0.068      0.088      0.136cm.; 

thus  the  effect  is  noticeable  even  with  relatively  small  iron  wire. 

Mutual  Induction 

115.  When  an  alternating  magnetic  field  of  force  includes  a 
secondary  electric  conductor,  it -generates  therein  an  e.m.f.  which 
produces  a  current,  and  thereby  consumes  energy  if  the  circuit 
of  the  secondary  conductor  is  closed. 


148         ALTERNATING-CURRENT  PHENOMENA 

Particular  cases  of  such  secondary  currents  are  the  eddy  or 
Foucault  currents  previously  discussed. 

Another  important  case  is  the  generation  of  secondary  e.m.fs. 
in  neighboring  circuits;  that  is,  the  interference  of  circuits  run- 
ning parallel  with  each  other. 

In  general,  it  is  preferable  to  consider  this  phenomenon  of 
mutual  induction  as  not  merely  producing  a  power  component 
and  a  wattless  component  of  e.m.f.  in  the  primary  conductor, 
but  to  consider  explicitly  both  the  secondary  and  the  primary 
circuit,  as  will  be  done  in  the  chapter  on  the  alternating-current 
transformer. 

Only  in  cases  where  the  energy  transferred  into  the  secondary 
circuit  constitutes  a  small  part  of  the  total  primary  energy,  as  in 
the  discussion  of  the  disturbance  caused  by  one  circuit  upon  a 
parallel  circuit,  may  the  effect  on  the  primary  circuit  be  con- 
sidered analogously  as  in  the  chapter  on  eddy  currents  by  the 
introduction  of  a  power  component,  representing  the  loss  of 
power,  and  a  wattless  component,  representing  the  decrease  of 
self-induction. 

Let 

x  =  2-n-fL  =  reactance  of  main  circuit;  that  is,  L  =  total  num- 
ber of  interlinkages  with  the  main  conductor,  of  the  lines  of 
magnetic  force  produced  by  unit  current  in  that  conductor; 

Xi  =  2irfLi  =  reactance  of  secondary  circuit;  that  is,  LI  = 
total  number  of  interlinkages  with  the  secondary  conductor,  of 
the  lines  of  magnetic  force  produced  by  unit  current  in  that  con- 
ductor; 

xm  —  2-n-fLi  =  mutual  inductive  reactance  of  the  circuits; 
that  is,  Lm  =  total  number  of  interlinkages  with  the  secondary 
conductor,  of  the  lines  of  magnetic  force  produced  by  unit  cur- 
rent in  the  main  conductor,  or  total  number  of  interlinkages 
with  the  main  conductor  of  the  lines  of  magnetic  force  produced 
by  unit  current  in  the  secondary  conductor. 
Obviously:  xm2  ^  xxi.1 

1  As  self -inductance  L,  LI,  the  total  flux  surrounding  the  conductor  is  here 
meant.  Usually  in  the  discussion  of  inductive  apparatus,  especially  of  trans- 
formers, as  the  self-inductance  of  circuit  is  denoted  that  part  of  the  mag- 
netic flux  which  surrounds  one  circuit  but  not  the  other  circuit;  and  as 
mutual  inductance  flux  which  passes  between  both  circuits.  Hence,  the 
total  self-inductance,  L,  is  in  this  case  equal  to  the  sum  of  the  self-induc- 
tance, LI,  and  mutual  inductance,  Lm. 

The  object  of  this  distinction  is  to  separate  the  wattless  part,  LI,  of  the 


FOUCAULT  OR  EDDY  CURRENTS  149 

Let  ri  =  resistance  of  secondary  circuit.     Then  the  imped- 
ance of  secondary  circuit  is 

Zi  =  n  +  jxi,  zi  =  \A*i2  +  zi2; 

e.m.f.  generated  in  the  secondary  circuit,  EI  =  —  jxml, 
where  /  =  primary  current.     Hence,  the  secondary  current  is 

T. 

J.    • 


and  the  e.m.f.  generated  in  the  primary  circuit  by  the  secondary 
current,  /i,  is 


or,  expanded, 

2 \      2    ~H         o   T [  f    /• 


Hence,  the  e.m.f.  consumed  thereby, 


E" 


/  =  (r  +  jx)L 


m 
2  ^_  —  ^  =  effective  resistance  of  mutual  inductance; 

"  ™*  3*       ^7*i 

o    m     o  =  effective  reactance  of  mutual  inductance. 


The  susceptance  of  mutual  inductance  is  negative,  or  of  opposite 
sign  from  the  reactance  of  self-inductance.  Or, 

Mutual  inductance  consumes  energy  and  decreases  the  self-in- 
ductance. 

For  the  calculation  of  the  mutual  inductance  between  circuits 
Lm,  see  "  Theoretical  Elements  of  Electrical  Engineering," 
4th  Ed. 

total  self  -inductance,  L,  from  that  part,  Lm,  which  represents  the  transfer  of 
e.m.f.  into  the  secondary  circuit,  since  the  action  of  these  two  components  is 
essentially  different. 

Thus,  in  alternating-current  transformers  it  is  customary  —  and  will  be 
done  later  in  this  book  —  to  denote  as  the  self-inductance,  L,  of  each  circuit 
only  that  part  of  the  magnetic  flux  produced  by  the  circuit  which  passes 
between  both  circuits,  and  thus  acts  in  "choking"  only,  but  not  in  trans- 
forming; while  the  flux  surrounding  both  circuits  is  called  the  mutual  induc- 
tance, or  useful  magnetic  flux. 

With  this  denotation,  in  transformers  the  mutual  inductance,  Lm,  is 
usually  very  much  greater  than  the  self-inductance,  L',  and  L/,  while,  if 
the  self-inductance,  L  and  LI,  represent  the  total  flux,  their  product  is  larger 
than  the  square  of  the  mutual  inductance,  Lm;  or 

LLi  ^  Lm2;  (L'  +  Lm)  (L/  +  Lm)  >  Lm*. 


CHAPTER  XIV 
DIELECTRIC  LOSSES 

Dielectric  Hysteresis 

116.  Just  as  magnetic  hysteresis  and  eddy  currents  give  a 
power  component  in  the  inductive  reactance,  as  "effective 
resistance,"  so  the  energy  losses  in  the  dielectric  lead  to  a  power 
component  in  the  condensive  reactance,  which  may  be  repre- 
sented by  an  "effective  resistance  of  dielectric  losses"  or  an 
"effective  conductance  of  dielectric  losses." 

In  the  alternating  magnetic  field,  power  is  consumed  by  mag- 
netic hysteresis.  This  is  proportional  to  the  frequency,  and  to 
the  1.6th  power  of  the  magnetic  density,  and  is  considerable, 
amounting  in  a  closed  magnetic  circuit  to  40  to  60  per  cent,  of  the 
total  volt-amperes. 

In  the  dielectric  field,  the  energy  losses  usually  are  very  much 
smaller,  rarely  amounting  to  more  than  a  few  per  cent.,  though 
they  may  at  high  temperature  in  cables  rise  as  high  as  40  to  60 
per  cent.  The  foremost  such  losses  are:  leakage,  that  is,  i2r  loss 
of  the  current  passing  by  conduction  (as  " dynamic  current") 
through  the  resistance  of  the  dielectric;  corona,  that  is,  losses 
due  to  a  partial  or  local  breakdown  of  the  electrostatic  field, 
and  dielectric  hysteresis  or  phenomena  of  similar  nature. 

It  is  doubtful  whether  a  true  dielectric  hysteresis,  that  is,  a 
molecular  dielectric  friction,  exists.  A  dielectric  loss,  propor- 
tional to  the  frequency  and  to  the  1.6th  power  of  the  dielectric 
field: 

P  =  n/D1-6 

has  been  observed  in  rotating  dielectric  fields,  but  is  so  small, 
•that  it  usually  is  overshadowed  by  the  other  losses. 

In  alternating  dielectric  fields  in  solid  materials,  such  as  in 
condensers,  coil  insulation,  etc.,  a  loss  is  commonly  observed 
which  gives  an  approximately  constant  power-factor  of  the  elec- 
tric energizing  circuit,  over  a  wide  range  of  voltage  and  of  fre- 
quency, from  less  than  a  fraction  of  1  per  cent,  up  to  a  few  per 
cent. 

150 


DIELECTRIC  LOSSES  151 

Constancy  of  the  power-factor  with  the  frequency,  means  that 
the  loss  is  proportional  to  the  frequency,  as  the  current  i,  and 
thus  the  volt-ampere  input,  ei,  are  proportional  to  the  frequency. 
Constancy  of  the  power-factor  with  the  voltage,  means  that  the 
loss  is  proportional  to  the  square  of  the  voltage,  as  the  current  i  is 
proportional  to  the  voltage,  and  the  volt-ampere  input  ei  thus 
proportional  to  the  square  of  the  voltage.  This  loss  thus  would 
be  approximated  by  the  expression  : 

P  =  rjfD* 

and  thus  seems  to  be  akin  to  magnetic  hysteresis,  except  that  at 
least  a  part  of  this  dielectric  loss  is  possibly  consumed  in  chemical 
and  mechanical  disintegration  of  the  insulating  material,  while 
the  magnetic  hysteresis  loss  is  entirely  converted  to  heat. 

Leakage 

117.  The  eddy  current  losses  in  the  magnetic  field  are  the  i*r 
loss  of  the  currents  flowing  in  the  magnetic  material,  and  as  such 
are  proportional  to  the  square  of  the  frequency  and  of  the  mag- 
netic density: 


where  7  =  conductivity  of  the  magnetic  material. 

This  expression  obviously  holds  only  as  long  as  the  m.m.f.  of 
the  eddy  currents  is  not  sufficient  to  appreciably  affect  the  mag- 
netic flux  distribution. 

As  corresponding  hereto  in  the  dielectric  field  may  be  con- 
sidered the  conduction  loss  through  the  resistance  of  the 
dielectric. 

In  a  homogeneous  dielectric  of  electric  conductivity  7  (usually 
very  low)  and  specific  capacity  or  permittivity  k,  if: 
I  =  thickness  of  the  dielectric, 

A  =  area  or  cross-section, 

e  =  impressed  alternating-current  voltage,  effective  value, 
the  dielectric  capacity  of  the  material  is: 

kA 


_ 

I 


and  the  capacity  susceptance: 


152         ALTERNATING-CURRENT  PHENOMENA 

hence  the  current  passing  through  the  dielectric  as  capacity 
current  or  "  displacement  current,"  is: 

2irfkA 
^o  =  eo  —  2  irfCe  =  —  -*  —  e 

The  conductance  of  the  dielectric  is: 


hence,  the  current,  conducted  through  the  dielectric,  or  leakage 
current: 


TA 

=  eg  =  -y 


thus,  the  total  current: 


here  the  j  denotes,  that  the  current  component  IQ  is  in  quadrature 
ahead  of  the  voltage  e. 

The  absolute  value  of  the  current  thus  is: 


and  the  power  consumption: 

. 
P  =  eii  = 


or,  since  the  dielectric  density  D  is  proportional  to  the  voltage 

/> 

gradient  j  and  the  permittivity: 

D  =     €k, 

(where  v  =  3  X  1010  =  velocity  of  light,  see  "  Theoretical  Ele- 
ments of  Electrical  Engineering.") 

Thus: 

P  =  -     V^jf- 
where 

V  =  Al  =  volume 

The  power-factor  then  is: 
P 


DIELECTRIC  LOSSES  153 

Or,  if,  as  usually  the  case,  the  conductivity  7  is  small  compared 
with  the  susceptivity  2  wfk : 

P  =  2tfk 

that  is,  the  power-factor  is  inverse  proportional  to  the  frequency. 

The  observation  of  leakage  losses  and  leakage  resistance  thus  is 
best  made  at  low  frequencies  or  at  direct-current  voltage. 

While,  however,  in  magnetic  materials  the  conductivity  7  is 
fairly  constant,  varying  only  with  the  temperature,  like  that  of 
all  metals,  the  very  low  conductivity  of  the  dielectric  is  often  not 
even  approximately  constant,  but  may  vary  with  the  tempera- 
ture, the  voltage,  etc.,  sometimes  by  many  thousand  per  cent. 

118.  While  in  a  homogeneous  dielectric  field,  the  leakage  cur- 
rent power  losses  are  independent  of  the  frequency  and  herein 
differ  from  the  magnetic  eddy  current  losses,  which  latter  are 
proportional  to  the  square  of  the  frequency,  in  non-homogene- 
ous dielectric  fields,  leakage  current  losses  may  depend  on  the 
frequency. 

As  an  instance,  let  us  consider  a  dielectric  consisting  of  two 
layers  of  different  constants,  for  instance,  a  layer  of  mica  and  a 
layer  of  varnished  cloth,  such  as  is  sometimes  used  in  high- 
voltage  armature  insulation. 

Let  71  =  electric  conductivity, 

ki  =  permittivity  or  specific  capacity, 
li  =  thickness  and, 
A  i  =  area  or  section 

of  the  first  layer  of  the  dielectric,  and 

72,   &2,   /2,   AZ 

the  corresponding  values  of  the  second  layer. 

It  is  then : 

yA 
g  =  —j-  =  electric  conductance 

kA 
C  =  -y  =  electrostatic  capacity    of    the    layer        . 

of  dielectric,  hence: 

2wfkA 
b  =  2irfC  =  — ^ —  =  capacity  susceptance,  and 


154         ALTERNATING-CURRENT  PHENOMENA 

Y  =  g  +  jb  =  admittance,  thus : 

Z  =y  =  r  —  jx  =  impedance,  where: 

r  =  -g  =  vector  resistance  (not  ohmic  resistance, 

but  energy  component  of  impedance,  (2) 

k          see  paragraph  89.) 
x  =  —2  =  vector  reactance,  and 

y  =  -y/fir2  +  fr2  =  absolute  admittance, 
(z  =  \/r2  +  x2  =  absolute  impedance.) 

If  then,  EI  =  potential  drop  across  the  first,  E2  =  potential 
drop  across  the  second  layer  of  dielectric, 

E  =  EI  -{-  E2  =  voltage  impressed  upon  the  dielectric.     (3) 
The  current  i,  which  traverses  the  dielectric,  partly  by  con- 
duction through  its  resistance,  partly  by  capacity  as  displace- 
ment current,  then  is  the  same  in  both  layers,  as  they  are  in 
series  in  the  dielectric  field,  and  it  is: 

EI  =  i(ri  -  jxi) 
E2  =  i(r2  —  jx2) 


and,  by  (3): 
or,  absolute : 


E 


r2)  - 


x2 


)  } 


r2) 


Thus,  the  current: 


V  (ri  +  r2)2  +  (xi  + 
the  apparent  power,  or  volt-ampere  input: 

e2 


Q  =  ei  = 


r2) 


the  power  consumed  in  the  dielectric  is: 
P  =  i»(fl  4.  r2) 

e2(ri  +  r2) 


(r, 


(x, 


and  the  power-factor: 


(4) 

(6) 
(6) 

(7) 
(8) 

(9) 
(10) 


Q       V(ri  +  r2)2  +  (Xl  +  *2)2 
119.  Let  us  consider  some  special  cases: 
(a)  If  the  conductivity,  71  and  y2)  of  the  two  layers  of  dielectric 


DIELECTRIC  LOSSES 


155 


is  so  small  that  the  conduction  current,  ge,  is  negligible  compared 
with  the  capacity  current,  2-jrfCe. 

In  this  case,  r\  and  r%  are  negligible  compared  with  xi  and  x2, 
and  it  is: 

e 


P_  c  y*  i  ~r  1 z 
/  —          I        _    V 


P  = 


(11) 


Substituting  now  for  the  ^impedance  quantities  Z  —  r  —  jx, 
which  have  no  direct  physical  meaning  in  the  dielectric  field,  the 
admittance  quantities  Y  =  g  +  jb,  which  have  the  physical 
meaning  that  g  is  the  effective  ohmic  conductance,  b  the  capacity 
susceptance,  it  is: 

g  negligible  compared  with  b  and  y,  and  b  =  y. 

Thus,  by  (2) : 

06162         2irfCiCze 
i    ^    =  ~r»     i    n  (12) 


_ 

~ 


hence  proportional  to  the  frequency  /: 


P  = 


4- 


(13) 


hence,  the  loss  of  power  by  current  leakage  in  the  dielectric  in  this 
case  is  independent  of  the  frequency. 

UQ  Oi  C/  2  C/ 1 

_  gi  6"i  +  g*  62  _  giC[  +  g*C~*  (14) 

6l  +   62  27T/(C1+C2) 

hence,  in  this  case  the  power-factor  is  inverse  proportional  to  the 
frequency. 

(6)  If  in  both  layers  the  leakage  current  is  large  compared  with 
the  capacity  current,  that  is,  2irfCe  negligible  compared  with  ge. 

In  this  case,  Xi  and  x%  are  negligible  compared  with  r\  and  r2, 
and: 


Q  = 


(15) 


156         ALTERNATING-CURRENT  PHENOMENA 

and  as  in  this  case  n  and  r2  are  the  effective  ohmic  resistance  of 
the  dielectric,  all  the  quantities  are  independent  of  the  frequency; 
that  is,  the  case  is  one  of  simple  conduction. 

120  (c)  If  in  the  first  layer  the  leakage  is  negligible  compared 
with  the  capacity  current,  but  is  not  negligible  in  the  second 
layer.  That  is,  in  a  two-layer  insulation,  one  layer  leaks  badly. 

Assuming  for  simplicity  that  the  two  layers  have  the  same 
capacity,  C  =  Ci  =  C2.  If  the  two  capacities  are  unequal,  the 
treatment  is  analogous,  but  merely  the  equations  somewhat  more 
complicated. 

Let  the  conductance  of  the  second  layer  =  g,  the  capacity 
susceptance  2  irfC  —  b. 

It  is  then: 

7*1  negligible  compared  with  the  other  quantities. 

g 


g*  + 
l 
b 
b 


(16) 


Substituting  these  values  in  equations  (7)  (8)  (9)  (10)  gives: 
e(g*  +  b2)  e(g*  +  (27r/C)2) 


V 


26\»  ~      I    g         4*/C\2     (17) 

7 


e*(g*  +  b2)  e*(g*  +  (27T/C)2) 

" 


As  seen,  in  this  case  current,  power  loss  and  power-factor  depend 
on  the  frequency,  but  in  a  more  complex  manner. 

With  changing  values  of  the  conductance  from  low  values, 
where  g  is  negligible  compared  with  the  other  terms,  but  the  other 

terms  negligible  compared  with  —  ,  up  to  high  conductivity,  where 

1  .  ** 

—  is  negligible,  but  the  terms  with  g  predominate,  the  current 

changes  from: 


DIELECTRIC  LOSSES  157 

low  g: 

i  =  jrfCe, 

proportional  to  the  frequency,  to: 
high  g: 

i  =  2TrfCe. 

Again  proportional  to  the  frequency,  but  twice  as  large,  and  at 
intermediate  values  of  g,  the  current  changes  more  rapidly  than 
proportional  to  the  frequency.  The  loss  o]  power  changes  from: 
low  g: 


or  independent  of  the  frequency,  to: 
high  g: 

p  = 


9 

or  proportional  to  the  square  of  the  frequency.     The  power-factor 
changes  from: 
low  g: 

' 


or  inverse  proportional  to  the  frequency,  to: 
high  g: 

27T/C 

P-    ~> 

or  proportional  to  the  frequency. 

And  over  a  considerable  range  of  intermediate  values  of  conduct- 
ance, g,  the  power-factor,  therefore,  remains  approximately  con- 
stant; or,  inversely,  with  changing  frequency  and  constant  g  and 
by  the  power-factor  changes  from  proportionality  with  the  fre- 
quency at  low  frequencies,  up  to  inverse  proportionality  at  high 
frequencies,  and  thereby  passes  through  a  maximum. 

The  value  of  g,  for  which  the  power-factor  in  equation  (19)  is 

a  maximum,  is  found  by  differentiating:  -j-  =  0,  as: 

g  =  2  V2  irfC  (20) 

and  this  maximum  power-factor  is  PQ  =  %. 

For  C2  >  Ci,  higher,  for  C2  <  Ci,  lower  values  of  power-factor 
maximum  result,  where  C2  is  the  leaky  dielectric. 


158 


ALTERNATING-CURRENT  PHENOMENA 


As  illustration,  Fig.  95  gives  the  values  of  power-factor,  p,  from 


g 


as  abscissae. 


equation  (19),  as  function  of  jr  = 

A  dielectric  circuit,  in  which  the  power-factor  decreases  with 
increasing  frequency,  for  instance,  is  that  of  the  capacity  of  the 
transmission  line;  a  dielectric  circuit,  in  which  the  power-factor 
increases  with  the  frequency,  is  that  of  the  aluminum-cell  light- 
ning arrester. 

121.  As  seen,  in  the  dielectric  circuit,  that  is,  in  insulators 
in  which  the  current  is  essentially  a  displacement  current,  the 


35 


10 


7 


.5       1.0     1.5      2.0     2.5     3.0     3.5      4.0     4.5     5.0     5.5 


6.5       7.0     7.5 


FIG.  95. 

relations  between  voltage,  current,  power,  phase  angle  and  power- 
factor  can  be  represented  by  the  same  symbolic  equations  as  the 
relations  between  voltage,  current,  power  and  power-factor  in 
metallic  conductors,  in  which  the  current  flow  is  dynamic,  by  the 
introduction  of  the  effective  admittance  of  the  dielectric  circuit,  or 
part  of  circuit: 

Y  =  g  +  jb, 

where  g  is  the  effective  conductance  of  the  dielectric  circuit,  or 
the  energy  component  of  the  admittance,  representing  the  energy 
consumption  by  leakage,  dielectric  hysteresis,  corona,  etc.,  and  b 
=  2  TT/C  is  the  capacity  susceptance.  Instead  of  the  admittance 
Y,  its  reciprocal,  the  impedance  Z  =  r  —  jx,  may  be  used. 

The  main  differences  between  the  dielectric  and  the  electro- 
dynamic  circuit  are: 

In  the  dielectric  circuit,  the  susceptance,  b,  is  positive,  the 
reactance,  x,  negative;  the  current  normally  leads  the  voltage, 


DIELECTRIC  LOSSES  159 

that  is,  capacity  effects  predominate  and  inductive  effects  are 
usually  absent. 

In  the  dynamic  circuit,  the  reactance,  x,  usually  is  positive, 
the  susceptance,  b,  negative;  the  current  usually  lags,  that  is, 
inductive  effects  predominate  and  capacity  effects  are  usually 
absent. 

In  the  dielectric  circuit,  the  admittance  terms,  Y  =  g  +  jb,  have 
a  physical  meaning  as  the  effective  conductance  and  the  capacity 
susceptance,  2  irfC,  but  the  impedance  terms,  Z  =  r  —  jx,  are  only 
derived  quantities,  without  direct  physical  meaning:  the  vector 
resistance,  r,  is  not  the  effective  ohmic  resistance  of  the  dielectric, 

-,  but  is  also  depending  on  the  capacity,  r  =    2  _,  ,  2,  and  the 

vector  reactance,  x,  is  not  the  condensive  reactance,  r-  ==  ~ — T^> 

^  ZTTJO 

but  also  depends  on  the  conductance,  x  =    2   ,   ,  2* 

In  the  dynamic  circuit,  the  impedance  terms,  Z  —  r  +  jx,  have 
a  direct  physical  meaning,  as  effective  ohmic  resistance,  r,  and 
as  self-inductive  reactance,  2irfL,  while  the  admittance  terms, 
Y  =  g  —  jb,  are  derived  quantities,  and  the  vector  conductance,  g, 
is  not  the  reciprocal  of  the  resistance,  r,  the  vector  susceptance,  6, 
not  the  reciprocal  of  the  reactance,  x,  as  discussed  in  preceding 
chapters. 

Physically,  the  most  prominent  difference  between  the  dielec- 
tric circuit  and  the  dynamic  circuit  is  that  for  the  displacement 
current  of  the  dielectric  circuit,  that  is,  for  the  electrostatic  flux, 
all  space  is  conducting,  while  for  the  dynamic  current,  most 
materials  are  practically  non-conductors,  and  the  dynamic  circuit 
thus  is  sharply  defined  in  the  extent  of  the  flow  of  the  current, 
while  the  dielectric  circuit  is  not.  The  dielectric  circuit  thus  is 
similar  to  the  magnetic  circuit;  for  the  magnetic  circuit  all  space 
is  conducting  also,  that  is,  can  carry  magnetic  flux.  An  unin- 
sulated submarine  electric  circuit  would  be  more  nearly  similar, 
in  the  distribution  of  current  flow,  to  the  dielectric  and  the  mag- 
netic circuit. 

In  the  electric  circuit,  the  conductor  through  which  the  cur- 
rent flows  is  generally  sharply  defined  and  of  a  uniform  section, 
which  is  small  compared  with  the  length,  and  the  conductor  thus 
can  be  approximated  as  a  linear  conductor,  that  is,  the  cur- 
rent distribution  throughout  the  conductor  section  assumed  as 
uniform.  With  the  dielectric  and  the  magnetic  circuit  this  is 


160         ALTERNATING-CURRENT  PHENOMENA 

rarely  the  case,  and  such  circuits  thus  have  to  be  investigated 
from  place  to  place  across  the  section  of  the  current  flow.  This 
brings  in  the  consideration  of  dielectric  current  density  or  dielec- 
tric flux  density,  and  corresponding  thereto  magnetic  flux  den- 
sity, as  commonly  used  terms,  while  dynamic  current  density, 
that  is,  current  per  unit  section  of  conductor,  is  far  less  frequently 
considered. 

Thus,  in  the  dielectric  circuit,  instead  of  admittance  Y  = 
g  +  jb,  commonly  the  admittance  per  unit  section  and  unit 
length  of  the  dielectric  circuit,  or  the  admittivity,  v  =  7  —  j0, 
has  to  be  considered,  where  7  =  conductivity  of  the  dielectric 
(or  effective  conductivity,  including  all  other  energy  losses),  and 
/8  =  2irfk  =  susceptivity,  where  k  =  permittivity  or  specific 
capacity  of  the  material. 

We  then  have: 


5    >       ™ 

4i<« 


122.  With  the  extended  industrial  use  of  very  high  voltage, 
the  explicit  study  of  the  dielectric  field  has  become  of  importance, 
and  it  is  not  safe  merely  to  consider  the  thickness  of  the  insulation 
in  relation  to  the  voltage  impressed  upon  it. 

In  an  ununiform  electric  conductor,  the  relation  of  the  voltage 
to  the  length  of  the  conductor  does  not  determine  whether  the 
conductor  is  safe  or  whether  locally,  due  to  small  cross-section  or 
high  resistivity,  unsafe  current  densities  may  cause  destructive 
heating,  but  the  adaptability  of  the  conductor  to  the  current 
carried  by  it  must  be  considered  throughout  its  entire  length. 
So  in  the  dielectric  field,  the  thickness  of  the  dielectric  may  be 
such  that  the  voltage  impressed  upon  it  may  give  a  very  safe 
average  voltage  gradient  or  average  dielectric  flux  density,  and 
the  dielectric  nevertheless  may  break  down,  due  to  local  concen- 
tration of  the  dielectric  flux  density  in  the  insulating  material. 
Thus,  for  instance,  in  the  dielectric  field  between  parallel  con- 
ductors, at  a  voltage  far  below  that  which  would  jump  from 
conductor  to  conductor,  locally  at  the  conductor  surface  the 
concentration  of  electrostatic  stress  exceeds  the  dielectric  strength 
of  air,  and  causes  it  to  break  down  as  corona.  In  solid  dielectrics, 
under  similar  conditions,  the  breakdown  due  to  local  over-stress 


DIELECTRIC  LOSSES  161 

often  may  change  the  flux  distribution  so  as  to  gradually  extend 
throughout  the  entire  dielectric,  until  puncture  results. 

Corona 

123.  —  In  the  magnetic  field,  with  increasing  magnetizing  force, 
/,  or  magnetic  field  intensity,  H,  the  magnetic  flux  density,  B, 
increases,  but  for  high  field  intensities  the  flux  density  ceases  to 
be  even  approximately  proportional  to  the  field  intensity,  and 
finally,  at  very  high  field  intensities,  H,  the  "metallic  magnetic 
induction,"  BQ  =  B  —  H,  reaches  a  finite  limiting  value,  which 
with  iron  is  not  far  from  BQ  —  20,000,  the  so-called  "  saturation 
value." 

In  the  dielectric  field,  with  increasing  voltage  gradient,  gr,  or 
dielectric  field  intensity,  K,  the  dielectric  flux  density,  D,  increases 
proportional  thereto,  until  a  finite  limiting  field  intensity,  K0)  or 
voltage  gradient,  g0,  is  reached,  beyond  which  the  dielectric  cannot 
be  stressed,  but  breaks  down  and  becomes  dynamically  conduct- 
ing, that  is,  punctures,  and  thereby  short-circuits  the  dielectric 
field. 

The  voltage  gradient,  gQ,  at  which  disruption  of  the  dielectric 
occurs  is  called  the  "disruptive  strength"  or  "dielectric 
strength"  of  the  dielectric.  With  air  at  atmospheric  pressure 
and  temperature,  it  is  go  —  30  kv.  per  centimeter.  Thus  under 
alternating  electric  stress,  air  punctures  at  21  kv.  effective  per 

centimeter  (  —7^  \  .     The  dielectric  strength  of  air  is  over  a  very 


wide  range  proportional  to  the  air  density,  and  thus  proportional 
to  the  barometric  pressure  and  inverse  proportional  to  the  abso- 
lute temperature.  Air  is  one  of  the  weakest  dielectrics,  and 
liquids  and  still  more  solids  show  far  higher  values  of  dielectric 
strength,  up  to  and  beyond  a  million  volts  per  centimeter. 

124.  If  then  in  a  uniform  dielectric  field,  such  as  that  between 
parallel  plates  A  and  B  as  shown  in  Fig.  96,  the  voltage  is  gradu- 
ally increased,  as  soon  as  the  voltage  maximum  reaches  a  gradi- 
ent of  0o  =  30  kv.  in  the  gap  between  the  metal  plates,  the  air 
in  this  gap  ceases  to  sustain  the  voltage,  a  spark  passes,  usually 
followed  by  the  arc,  and  the  potential  difference  across  this  gap 
drops  from  g0l  —  where  I  is  the  distance  between  the  metal  plates 
A  and  B  —  to  practically  nothing,  and  the  electric  circuit  thereby 

ceases  to  include  a  dielectric  field. 
11 


162         ALTERNATING-CURRENT  PHENOMENA 

Assuming  now  that  the  gap  between  the  metal  plates  does  not 
contain  a  homogeneous  dielectric,  but  one  consisting  of  several 
layers  of  different  dielectric  strength  and  different  permittivity. 
For  instance,  we  put  two  glass  plates,  a  and  b}  of  thickness  10  into 
the  gap,  as  shown  in  Fig.  97,  thereby  leaving  an  air  space,  c,  of 
I  —  2  1Q.  The  dielectric  flux  density  in  the  field  is  still  uniform 


\   / 

Al        IB 


/   V 


FIG.  96. 


FIG.  97. 


throughout  the  field  section,  but  the  voltage  gradient  in  the 

different  layers,  a,  b  and  c,  is  not  the  same,  is  not  the  average  gra- 

g 
dient,  g  =  -y  ,  of  the  gap,  but  is  inverse  proportional  to  the  permit- 

tivities : 

1          . 


where  A;0  is  the  permittivity  of  the  layers,  a  and  6,  k\  the  permit- 
tivity of  the  layer  c  (  =  1,  if  this  layer  is  air).  The  potential 
drop  across  a  and  b  thus  is  l0gQ)  across  c  is  (I  —  2  10)g^  and  the 
total  voltage  thus: 

e  =  2  IQ  gQ  +  (I  —  2  Wfifi, 


DIELECTRIC  LOSSES  163 

g\k\ 
or,  substituting  gQ  =  -r—  gives: 


hence: 

e  ek( 


-       i     2  !.(*,-*„)+  ft 

and 


Depending  on  the  values  of  k\  and  fco,  either  g$  or  <7i  may  be  higher 
than  the  average  gradient 

e 

9~r 

To  illustrate  on  a  numerical  instance: 

Let  the  distance  between  the  metal  plates  A  and  B  be  I  =  1  cm. 
With  nothing  but  air  at  atmospheric  pressure  and  temperature 
between  the  plates,  the  gap  would  break  down  by  a  spark  dis- 
charge, and  short-circuit  the  circuit  of  Fig.  96;  at  e  =  30  kv. 
maximum,  and  at  e  =  25  kv.,  no  discharge  would  occur. 

Assuming  now  two  glass  plates,  a  and  b,  each  of  0.3  cm.  thick- 
ness and  permittivity  /c0  =  4,  were  inserted,  leaving  an  air-gap 
of  0.4  cm.  of  permittivity  ki  =  1.  At  e  =  25  kv.  the  gradients 
thus  would  be,  by  above  equation: 

In  the  glass  plates: 

g\  =  8.4  kv.  per  cm. 

In  the  air-gap: 

go  =  35.7  kv.  per  cm. 

The  air  would  thus  be  stressed  beyond  its  dielectric  strength, 
and  would  break  down  by  spark  discharge.  This  would  drop 
the  gradient  in  the  air  down  to  practically  g'o  —  0,  and  the 
gradient  in  the  glass  plates  thus  would  become  : 

/         ^"  «  i 

jfi  =  Q-Z  =  41.7  kv.  per  cm. 

Thus  the  insertion  of  the  glass  plates  would  cause  the  air-gap 
to  break  down.  The  dynamic  current  which  flows  through  the 
air-gap  in  this  case  would  not  be  the  short-circuit  current  of  the 


164         ALTERNATING-CURRENT  PHENOMENA 

electric  circuit,  as  would  be  the  case  in  the  absence  of  the  glass 
plates  but  it  would  merely  be  the  capacity  current  of  the  glass 
plates;  and  it  would  not  be  followed  by  the  arc,  but  passes  as  a 
uniform  bluish  glow  discharge,  or  as  pink  streamers — corona. 

125.  If  the  dielectric  field  is  not  uniform,  but  varying  in  density 
as,  for  instance,  the  field  between  two  spheres  or  the  field  between 
two  parallel  wires,  then  with  increasing  voltage  the  breakdown 
gradient  will  not  be  reached  simultaneously  throughout  the  en- 
tire field,  as  in  a  uniform  field,  but  it  is  first  reached  in  the  denser 
portion  of  the  field — at  the  surface  of  the  spheres  or  parallel  wires, 
where  the  lines  of  dielectric  force  converge.  Thus  the  dielectric 
will  first  break  down  at  the  denser  portion  of  the  field,  and  short- 
circuit  these  portions  by  the  flow  of  dynamic  current.  This, 
however,  changes  the  voltage  gradient  in  the  rest  of  the  field, 
and  may  raise  it  so  as  to  break  down  the  entire  field,  or  it  may  not 
do  so. 


FIG.  98. 


O 


FIG.  99. 

For  instance,  in  the  dielectric  field  between  two  spheres  at 
distance  I  from  each  other,  as  shown  in  Figs.  98  and  99,  with  in- 
creasing potential  difference,  e,  finally  the  breakdown  gradient  of 
the  air,  gQ  =  30  kv.  =  cm.,  is  reached  at  the  surface  of  the  spheres, 
and  up  to  a  certain  distance  6  beyond  it,  and  in  this  space  d  the 
air  breaks  down,  becomes  conducting,  and  the  space  up  to  the 
distance  d  is  filled  with  corona.  As  the  result,  the  conducting 
terminals  of  the  dielectric  field  are  not  the  original  spheres,  but  the 
entire  space  filled  by  the  corona,  that  is,  the  terminals  are  in- 
creased in  size,  and  the  convergency  of  the  dielectric  flux  lines, 
that  is,  the  voltage  gradient  at  the  effective  terminals,  is  reduced. 
At  the  same  time  the  gap  between  the  effective  terminals  is  re- 
duced by  25,  and  the  average  voltage  gradient  thereby  increased. 


DIELECTRIC  LOSSES  165 

If  the  latter  effect  is  greater — as  is  the  case  with  large  spheres  at 
short  distance  from  each  other — the  air  becomes  over-stressed  at 
the  edge  d  of  the  corona  formed  by  the  original  field,  the  corona 
spreads  farther,  and  so  on,  until  the  entire  field  breaks  down,  that 
is,  no  stable  corona  forms,  but  immediate  disruptive  discharge. 
Inversely,  with  small  spheres  at  considerable  distance  from  each 
other,  the  formation  of  corona  very  soon  increases  the  size  of  the 
effective  terminals  so  as  to  bring  the  voltage  gradient  at  the  edge 
of  the  corona  down  to  the  disruptive  gradient,  g0,  and  the  corona 
spreads  no  farther.  In  this  case  then,  with  increasing  voltage, 
at  a  certain  voltage,  e0,  corona  begins  to  form  at  the  terminals, 
first  as  bluish  glow,  then  as  violet  streamers,  which  spread  farther 
and  farther  with  increasing  voltage,  until  finally  the  disruptive 
spark  passes'  between  the  terminals.  In  this  case,  corona  pre- 
cedes the  disruptive  discharge. 

Experience  shows  that  the  voltage,  ev,  at  which  corona  begins 
at  the  surface  is  not  the  voltage  at  which  the  breakdown  gradient 
of  air,  gQ  =  30,  is  reached  at  the  sphere  surface,  but  ev  is  the  vol- 
tage at  which  the  breakdown  gradient,  go,  has  extended  up  to  a 
certain  small  but  definite  distance  the  " energy  distance"  from 
the  spheres.  That  is,  dielectric  breakdown  of  the  air  requires  a 
finite  volume  of  over-stressed  air,  that  is,  a  finite  amount  of  di- 
electric energy.  As  the  result,  when  corona  begins,  the  gradient 
at  the  terminal  surface,  gv,  is  higher  than  the  breakdown  gradi- 
ent, 0o,  the  more  so  the  more  the  flux  lines  converge,  that  is, 
the  smaller  the  spheres  (or  parallel  wires)  are. 

126.  With  the  development  of  high-voltage  transmission  at 
100  kv.  and  over,  the  electrical  industry  has  entered  the  range  of 
voltage,  where  corona  appears  on  parallel  wires  of  sizes  such 
as  are  industrially  used.  Such  corona  consumes  power,  and 
thereby  introduces  an  energy  component  into  the  expression  of 
the  line  capacity,  a  corona  conductance. 

The  power  consumption  by  the  corona  is  approximately 
proportional  to  the  frequency,  its  power  factor  therefore  inde- 
pendent of  the  frequency. 

The  power  consumption  by  the  corona  is  proportional  to  the 
square  of  the  excess  voltage  over  that  voltage,  e0,  which  brings  the 
dielectric  field  at  the  conductor  surface  up  to  the  breakdown 
gradient,  g0. 

However,  corona  does  not  yet  appear  at  the  voltage,  e0,  which 
produces  the  breakdown  gradient,  g0,  at  the  conductor  surface, 


166         ALTERNATING-CURRENT  PHENOMENA 

but  at  the  higher  voltage,  ev,  which  has  extended  the  breakdown 
gradient  by  the  energy  distance  from  the  conductor  surface. 
Then  the  corona  power  begins  with  a  finite  value,  and  in  the 
range  between  e0  and  ev  it  is  indefinite,  depending  on  the  surface 
condition  of  the  conductor. 

The  equations  of  the  power  consumption  by  corona  in  parallel 
conductors  are: 


where  : 

P  =  power  loss  in  kilowatts  per  kilometer  length  of  single- 

line  conductor; 
e  =  effective  value  of  the  voltage  between  the  line  conductor 

and  neutral  in  kilo  volts;1 
/  =  frequency; 
c  =  25; 

and  a  is  given  by  the  equation  : 

A  rr 

a  =  T\/- 
5  \  s 

where  : 

r  =  radius  of  conductor  in  centimeters; 

s  =  distance  between  conductor  and  return   conductor  in 

centimeters; 

6  =  density  of  the  air,  referred  to  25°C.  and  76  cm.  barometer; 
A  =  241; 

and: 

e0  =  effective  disruptive  critical  voltage  to  neutral,  given  in 
kilovolts  by  the  equation  (natural  logarithm) 

o 

CQ  =  m0gQ  5r  log  -• 

where  : 

0o  =  21.1  kv.  per  centimeter  effective  =  breakdown  gradient 

of  air; 
Wo  =  surface  constant  of  the  conductor. 

It  is: 

mo  =  1  for  perfectly  smooth  polished  wire; 

Wo  =  0.98  to  0.93  for  roughened  or  weathered  wire; 

1  =  }4  the  voltage  between  the  conductors  in  a  single-phase  circuit,  1/V3 
times  the  voltage  between  the  conductors  in  a  three-phase  circuit. 


DIELECTRIC  LOSSES  167 

decreasing  to: 

wo  =  0.87  to  0.83  for7-strand  cable  (r  being  the  outer  radius  of 
the  cable).1 

Materially  higher  losses  occur  in  snow  storms  and  rain. 

For  further  discussion  of  the  dielectric  field  and  the  power 
losses  in  it,  see  F.  W.  Peek's  "Dielectric  Phenomena  in  High- 
voltage  Engineering." 

1  "Dielectric  Phenomena  in  High-voltage  Engineering,"  by  F.  W.  Peek, 
Jr.,  page  200. 


CHAPTER  XV 

DISTRIBUTED   CAPACITY,   INDUCTANCE,   RESISTANCE, 
AND  LEAKAGE 

127.  In  the  foregoing,  the  phenomena  causing  loss  of  energy 
in  an  alternating-current  circuit  have  been  discussed;  and  it  has 
been  shown  that  the  mutual  relation  between  current  and  e.m.f. 
can  be  expressed  by  two  of  the  four  constants: 

power  component  of  e.m.f.,  in  phase  with  current,  and  =  current 

X  effective  resistance,  or  r; 
reactive  component  of  e.m.f.,  in  quadrature  with  current,  and  = 

current  X  effective  reactance,  or  x; 
power  component  of  current,  in  phase  with  e.m.f.,  and  =  e.m.f. 

X  effective  conductance,  or  0; 
reactive  component  of  current,  in  quadrature  with  e.m.f.,  and  = 

e.m.f.  X  effective  susceptance,  or  b. 

In  many  cases  the  exact  calculation  of  the  quantities,  r,  x,  g,  6, 
is  not  possible  in  the  present  state  of  the  art. 

In  general,  r,  x,  g,  6,  are  not  constants  of  the  circuit,  but  depend 
— besides  upon  the  frequency — more  or  less  upon  e.m.f.,  current, 
etc.  Thus,  in  each  particular  case  it  becomes  necessary  to  dis- 
cuss the  variation  of  r,  x,  g,  6,  or  to  determine  whether,  and 
through  what  range,  they  can  be  assumed  as  constant. 

In  what  follows,  the  quantities  r,  x,  g,  6,  will  always  be  consid- 
ered as  the  coefficients  of  the  power  and  reactive  components  of 
current  and  e.m.f. — that  is,  as  the  effective  quantities — so  that 
the  results  are  directly  applicable  to  the  general  electric  circuit 
containing  iron  and  dielectric  losses. 

Introducing  now,  in  Chapters  VIII,  to  XI,  instead  of  "ohmic 
resistance,"  the  term  "effective  resistance,"  etc.,  as  discussed 
in  the  preceding  chapter,  the  results  apply  also — within  the  range 
discussed  in  the  preceding  chapter — to  circuits  containing  iron 
and  other  materials  producing  energy  losses  outside  of  the  electric 
conductor. 

128.  As  far  as  capacity  has  been  considered  in  the  foregoing 
chapters,  the  assumption  has  been  made  that  the  condenser  or 

168 


DISTRIBUTED  CAPACITY  169 

other  source  of  negative  reactance  is  shunted  across  the  circuit  at 
a  definite  point.  In  many  cases,  however,  the  condensive  react- 
ance is  distributed  over  the  whole  length  of  the  conductor,  so 
that  the  circuit  can  be  considered  as  shunted  by  an  infinite  num- 
ber of  infinitely  small  condensers  infinitely  near  together,  as 
diagrammatically  shown  in  Fig.  100. 


iiiiiiiiiiiiii 11 iiiii 

TTTTTTTTTTTTTTTTTTTTT 


FIG.  100. 

In  this  case  the  intensity  as  well  as  phase  of  the  current,  and 
consequently  of  the  counter  e.m.f.  of  inductive  reactance  and 
resistance,  vary  from  point  to  point;  and  it  is  no  longer  possible 
to  treat  the  circuit  in  the  usual  manner  by  the  vector  diagram. 

This  phenomenon  is  especially  noticeable  in  long-distance  lines, 
in  underground  cables,  and  to  a  certain  degree  in  the  high-poten- 
tial coils  of  alternating-current  transformers  for  very  high  vol- 
tage and  also  in  high  frequency  circuits.  It  has  the  effect  that  not 
only  the  e.m.fs.,  but  also  the  currents,  at  the  beginning,  end,  and 
different  points  of  the  conductor,  are  different  in  intensity  and  in 
phase. 

Where  the  capacity  effect  of  the  line  is  small,  it  may  with 
sufficient  approximation  be  represented  by  one  condenser  of  the 
same  capacity  as  the  line,  shunted  across  the  line  at  its  middle. 
Frequently  it  makes  no  difference  either,  whether  this  condenser  is 
considered  as  connected  across  the  line  at  the  generator  end,  or 
at  the  receiver  end,  or  at  the  middle. 

A  better  approximation  is  to  consider  the  line  as  shunted  at 
the  generator  and  at  the  motor  end,  by  two  condensers  of  one- 
sixth  the  line  capacity  each,  and  in  the  middle  by  a  condenser 
of  two-thirds  the  line  capacity.  This  approximation,  based  on 
Simpson's  rule,  assumes  the  variation  of  the  electric  quantities 
in  the  line  as  parabolic.  If,  however,  the  capacity  of  the  line  is 
considerable,  and  the  condenser  current  is  of  the  same  magnitude 
as  the  main  current,  such  an  approximation  is  not  permissible, 
but  each  line  element  has  to  be  considered  as  an  infinitely  small 
condenser,  and  the  differential  equations  based  thereon  integrated. 
Or  the  phenomena  occurring  in  the  circuit  can  be  investigated 
graphically  by  the  method  given  in  Chapter  VI,  §39,  by  dividing 
the  circuit  into  a  sufficiently  large  number  of  sections  or  line 


170         ALTERNATING-CURRENT  PHENOMENA 

elements,  and  then  passing  from  line  element  to  line  element,  to 
construct  the  topographic  circuit  characteristics. 

129.  It  is  thus  desirable  to  first  investigate  the  limits  of  appli- 
cability of  the  approximate  representation  of  the  line  by  one  or 
by  three  condensers. 

Assuming,  for  instance,  that  the  line  conductors  are  of  1  cm. 
diameter,  and  at  a  distance  from  each  other  of  50  cm.,  and  that 
the  length  of  transmission  is  50  km.,  we  get  the  capacity  of  the 
transmission  line  from  the  formula — 

C  =  1.11  X  10~6  kl  -T-  4  log,  2^  microfarads, 

where 

k  =  dielectric  constant  of  the  surrounding  medium  =  1  in  air; 

I  =  length  of  conductor  =  5  X  106  cm.; 

d  =  distance  of  conductors  from  each  other  =  50  cm.; 

d  =  diameter  of  conductor  =  1  cm. 
Hence  C  =  0.3  microfarad, 

the  condensive  reactance  is  x  =  ^ — 77?  ohms, 

Z  7T/O 

where/  =  frequency;  hence  at/  =  60  cycles, 

x  =  8,900  ohms; 

and  the  charging  current  of  the  line,  at  E  =  20,000  volts,  be- 
comes, 

E 

IQ  =  —  =  2.25  amp. 
x 

The  resistance  of  100  km.  of  wire  of  1  cm.  diameter  is  22  ohms; 
therefore,  at  10  per  cent.  =  2,000  volts  loss  in  the  line,  the  main 
current  transmitted  over  the  line  is 

7  2>°°°  01 

/  =  -w  =  91  amp. 

representing  about  1,800  kw. 

In  this  case,  the  condenser  current  thus  amounts  to  less  than 
2.5  per  cent.,  and  hence  can  still  be  represented  by  the  approxi- 
mation of  one  condenser  shunted  across  the  line. 

If  the  length  of  transmission  is  150  km.,  and  the  voltage, 
30,000, 

condensive  reactance  at  60  cycles,      x  ='  2,970  ohms; 

charging  current,  IQ  =  10.1  amp.; 

line  resistance,  r  =  66  ohms; 

main  current  at  10  per  cent,  loss,        /  =  45.5  amp. 


DISTRIBUTED  CAPACITY  111 

The  condenser  current  is  thus  about  22  per  cent,  of  the  main 
current,  and  the  approximate  calculation  of  the  effect  of  line 
capacity  still  fairly  accurate. 

At  300  km.  length  of  transmission  it  will,  at  10  per  cent, 
loss  and  with  the  same  size  of  conductor,  rise  to  nearly  90  per 
cent,  of  the  main  current,  thus  making  a  more  explicit  investiga- 
tion of  the  phenomena  in  the  line  necessary. 

In  many  cases  of  practical  engineering,  however,  the  capacity 
effect  is  small  enough  to  be  represented  by  the  approximation 
of  one;  or,  three  condensers  shunted  across  the  line. 

130.  (A)  Line  capacity  represented  by  one  condenser  shunted 
across  middle  of  line. 

Let 

Y  =  g  —  jb  =  admittance  of  receiving  circuit; 
Z  =  r  +  jx  =  impedance  of  line ; 
bc  =  condenser  susceptance  of  line. 


li 

J 

Ti 

li° 

RF—                                             i  g 

Y                                                                                 LI 

FIG.  101. 

Denoting  in  Fig.  101. 

the  e.m.f.,  and  current  in  receiving  circuit  by  E,  7, 
the  e.m.f.  at  middle  of  line  by  E', 
the  e.m.f.,  and  current  at  generator  by  EQ,  /oj 
we  have, 

1  =  E(g-fl>); 


.(r+jx)  (g-Jb)} 
2  I 

70  =  7  +  jbcE' 


wt«    ,   (r+jx)  (g-jb)       (r+jx)  (g-jb) 
=  ^11  +  ~^~  ~T~ 

,  jbc(r  +  jx)  (r  +  jx)2  fa  -  j6)  \ 

2          ^J°c  4  P 


172         ALTERNATING-CURRENT  PHENOMENA 
or,  expanding, 

/.  =  E[  {g  +  b^(rb  -  xg)]  -  j  [(b  -  6.)  -  ^(rg  +  xb)]  }  • 
1  +  (r  +  jx)  (g-jb)  +J-     (r+jx) 


=  B  {  1  +  (r  +  jx)  (g  -  jb  +  ]^)  +  ^(r+jxY  (g  -JK)  \  - 


131.  Distributed  condensive  reactance,  inductive  reactance,  leak- 
age, and  resistance. 

In  some  cases,  especially  in  very  long  circuits,  as  in  lines 
conveying  alternating-power  currents  at  high  potential  over 
extremely  long  distances  by  overhead  conductors  or  under- 
ground cables,  or  with  very  feeble  currents  at  extremely  high 
frequency,  such  as  telephone  currents,  the  consideration  of  the 
line  resistance  —  which  consumes  e.m.fs.  in  phase  with  the  current 
—  and  of  the  line  reactance  —  which  consumes  e.m.fs.  in  quadrature 
with  the  current  —  is  not  sufficient  for  the  explanation  of  the 
phenomena  taking  place  in  the  line,  but  several  other  factors 
have  to  be  taken  into  account. 

In  long  lines,  especially  at  high  potentials,  the  electrostatic 
capacity  of  the  line  is  sufficient  to  consume  noticeable  currents. 
The  charging  current  of  the  line  condenser  is  proportional  to  the 
difference  of  potential,  and  is  one-fourth  period  ahead  of  the 
e.m.f.  Hence,  it  will  either  increase  or  decrease  the  main 
current,  according  to  the  relative  phase  of  the  main  current  and 
the  e.m.f. 

As  a  consequence,  the  current  changes  in  intensity  as  well 
as  in  phase,  in  the  line  from  point  to  point;  and  the  e.m.f.  con- 
sumed by  the  resistance  and  inductive  reactance  therefore  also 
changes  in  phase  and  intensity  from  point  to  point,  being 
dependent  upon  the  current. 

Since  no  insulator  has  an  infinite  resistance,  and  as  at  high 
potentials  not  only  leakage,  but  even  direct  escape  of  electricity 
into  the  air,  takes  place  by  corona,  we  have  to  recognize  the 
existence  of  a  current  approximately  proportional  and  in  phase 
with  the  e.m.f.  of  the  line.  This  current  represents  consumption 
of  power,  and  is,  therefore,  analogous  to  the  e.m.f.  consumed 
by  resistance,  while  the  condenser  current  and  the  e.m.f.  of  self- 
induction  are  wattless  or  reactive. 


DISTRIBUTED  CAPACITY  173 

Furthermore,  the  alternating  current  in  the  line  produces  in  all 
neighboring  conductors  secondary  currents,  which  react  upon 
the  primary  current,  and  thereby  introduce  e.m.fs.  of  mutual 
inductance  into  the  primary  circuit.  Mutual  inductance  is 
neither  in  phase  nor  in  quadrature  with  the  current,  and  can 
therefore  be  resolved  into  a  power  component  of  mutual  induct- 
ance in  phase  with  the  current,  which  acts  as  an  increase  of 
resistance,  and  into  a  reactive  component  in  quadrature  with  the 
current,  which  decreases  the  self-inductance. 

This  mutual  inductance  is  not  always  negligible,  as,  for  in- 
stance, its  disturbing  influence  in  telephone  circuits  shows. 

The  alternating  voltage  of  the  line  induces,  by  electrostatic 
influence,  electric  charges  in  neighboring  conductors  outside  of 
the  circuit,  which  retain  corresponding  opposite  charges  on  the 
line  wires.  This  electrostatic  influence  requires  a  current  pro- 
portional to  the  e.m.f.  and  consisting  of  a  power  component,  in 
phase  with  the  e.m.f.,  and  a  reactive  component,  in  quadrature 
thereto. 

The  alternating  electromagnetic  field  of  force  set  up  by  the 
line  current  produces  in  some  materials  a  loss  of  energy  by 
magnetic  hysteresis,  or  an  expenditure  of  e.m.f.  in  phase  with 
the  current,  which  acts  as  an  increase  of  resistance.  This 
electromagnetic  hysteretic  loss  may  take  place  in  the  con- 
ductor proper  if  iron  wires  are  used,  and  will  then  be  very  serious 
at  high  frequencies,  such  as  those  of  telephone  currents. 

The  effect  of  eddy  currents  has  already  been  referred  to  under 
"mutual  inductive  reactance,"  of  which  it  is  a  power  component. 

The  alternating  electrostatic  field  of  force  expends  energy  in 
dielectrics  by  corona  and  dielectric  hysteresis.  In  concentric 
cables,  where  the  electrostatic  gradient  in  the  dielectric  is  com- 
paratively large,  the  dielectric  losses  may  at  high  potentials 
consume  appreciable  amounts  of  energy.  The  dielectric  loss 
appears  in  the  circuit  as  consumption  of  a  current,  whose  com- 
ponent in  phase  with  the  e  m.f.  is  the  dielectric  power  current, 
which  may  be  considered  as  the  power  component  of  the  capacity 
current. 

Besides  this,  there  is  the  increase  of  ohmic  resistance  due  to 
unequal  distribution  of  current,  which,  however,  is  usually  not 
large  enough  to  be  noticeable. 

Furthermore,  the  electric  field  of  the  conductor  progresses 
with  a  finite  velocity,  the  velocity  of  light,  hence  lags  behind 


174         ALTERNATING-CURRENT  PHENOMENA 

the  flow  of  power  in  the  conductor,  and  so  also  introduces 
power  components,  depending  on  current  as  well  as  on  potential 
difference. 

132.  This  gives,  as  the  most  general  case,  and  per  unit  length 
of  line : 

e.m.fs.  consumed  in  phase  with  the  current,  I,  and  =  rl,  repre- 
senting consumption  of  power,  and  due  to: 
Resistance,    and    its    increase    by    unequal    current    distri- 
bution;   to   the    power   component    of   mutual   inductive 
reactance  or  to  induced  currents;  to  the  power  component 
of  self-inductive  reactance  or  to  electromagnetic  hysteresis, 
and  to  radiation. 
e.m.fs.  consumed  in  quadrature  with  the  current,  I,  and  =  xl, 

wattless,  and  due  to: 
Self -inductance,  and  mutual  inductance. 
Currents  consumed  in  phase  with  the  e.m.f.,  E,  and    =  g  E, 

representing  consumption  of  power,  and  due  to: 
Leakage  through  the  insulating  material,  including  silent 
discharge  and  corona;  power  component  of  electrostatic 
influence;    power    component    of    capacity    or    dielectric 
hysteresis,  and  to  radiation. 
Currents  consumed  in  quadrature  to  the  e.m.f.,  E,  and   =  bE, 

being  wattless,  and  due  to: 
Capacity  and  electrostatic  influence. 
Hence  we  get  four  constants: 
Effective  resistance,  r, 
Effective  reactance,  z, 
Effective  conductance,  g, 
Effective  susceptance,  —  b, 

per  unit  length  of  line,  which  represents  the  coefficients,  per  unit 
lenght  of  line,  of 

e.m.f.  consumed  in  phase  with  current; 
e.m.f.  consumed  in  quadrature  with  current; 
current  consumed  in  phase  with  e.m.f.; 
current  consumed  in  quadrature  with  e.m.f.; 
or, 

Z  =  r  +  jx, 
Y  =  g+jb, 
and,  absolute, 

z  = 

y  = 


DISTRIBUTED  CAPACITY  175 

The  complete  investigation  of  a  circuit  or  line  contain- 
ing distributed  capacity,  inductive  reactance,  resistance,  etc., 
leads  to  functions  which  are  products  of  exponential  and  of 
trigonometric  functions.  That  is,  the  current  and  potential 
difference  along  the  line,  Z,  are  given  by  expressions  of  the  form  : 

e+al(A  cos  #  +  B  sin  01). 

Such  functions  of  the  distance,  I,  or  position  on  the  line, 
while  alternating  in  time,  differ  from  the  true  alternating  waves 
in  that  the  intensities  of  successive  half-waves  progressively 
increase  or  decrease  with  the  distance.  Such  functions  are  called 
oscillating  waves,  and,  as  such,  are  beyond  the  scope  of  this 
book,  but  are  more  fully  treated  in  "  Theory  and  Calculation 
of  Transient  Electric  Phenomena  an,d  Oscillations/'  Section  III. 
There  also  will  be  found  the  discussion  of  the  phenomena  of 
distributed  capacity  in  high-potential  transformer  windings,  the 
effect  of  the  finite  velocity  of  propagation  of  the  electric  field,  etc. 

For  most  purposes,  however,  in  calculating  long-distance 
transmission  lines  and  other  circuits  of  distributed  constants, 
the  following  approximate  solutions  of  the  general  differential 
equation  of  the  circuit  offers  sufficient  exactness. 

133.  The  impedance  of  an  element,  dl,  of  the  line  is: 

Zdl 
and  the  voltage,  dE,  consumed  by  the  current,  /,  in  this  line  ele- 

ment  dl:  dE  =  Zldl 

The  admittance  of  the  line  element,  dl,  is: 

Ydl 

hence  the  current,  dl,  consumed  by  the  voltage,  dE,  of  this  line 
element  <fi:  '  a  -  YEdl 

This  gives  the  two  equations  of  the  transmission  line: 


Differentiating  the  first  equation,  and  substituting  therein  the 
second,  gives:  • 


176         ALTERNATING-CURRENT  PHENOMENA 
and  from  the  first  equation  follows: 

/    =    7^    ~TT  (2) 

£j      Oil 

Equation  (1)  is  integrated  by: 

E  =  Aem  (3) 

and,  substituting  (3)  in  (1),  gives: 

B2  =  ZY 
hence: 

B  =  +  VZY  and  -  \fZY 

There  exist  thus  two  values  of  B,  which  make  (3)  a  solution  of 
(1),  and  the  most  general  solution,  therefore,  is: 

E  =  A i  e  +A2  e  (4) 

Substituting  (4)  in  (2)  gives: 

+VzTi  -VzYi] 

Af.  A  £  f^ 

i  e  -  /1 2  c  V°/ 

where  I  is  counted  from  some  point  of  the  line  as  starting  point, 
for  instance,  from  the  step-down  end  as  I  =  0. 

If  then: 

EQ  =  voltage  at  step-down  end  of  the  line, 
70  =  current  at  step-down  end, 
it  is,  for:  I  =  0; 

EQ   =   Ai  +  A2 


hence: 

(6) 


and,  substituting  (6)  into  (5): 


7  -  T 
/-/ 


+  VZYI       -  V^YI          r&        + 

+  6  -L.        Z    T   € 

~  "  ° 


*  A    '  2  C7) 

-Vz'ri  r=        +VZYI       -VZYI 


DISTRIBUTED  CAPACITY  177 

Substituting  in  (7)  for  the  exponential  function  the  infinite 
series : 


l£  • 

gives: 


(8) 


134.  If  then:  /  =  10  is  the  total  length  of  line,  and 
ZQ  =  loZ   =  total  line  impedance, 
Y0  =  10Y  =  total  line  admittance, 

the  equations  of  voltage  EI  and  current  /i    at   the   end    1Q  of 
the  line  are  given  by  substituting  I  =  1Q  into  equations  (8),  as: 


(9) 


2 

Since  Z0  is  the  line  impedance,  and  thus  Z0I  the  impedance 

voltage,  — ~  is  the  impedance  voltage,  as  fraction  of  the  total 

voltage.     Since  F0  is  the  line  admittance,  Y0E  is  the  charging 

V  7? 

current,  and  — j—  the  charging  current  as  fraction  of  the  total 

current.     The  product  of  these  two  fractions  is: 


v  ,  \/ 
E    A     / 

Z0F0  thus  is  the  product  of  impedance  voltage  and  charging 
current  of  the  line,  expressed  as  fraction  of  total  voltage  and  total 
current,  respectively,  hence  is  a  small  quantity,  and  its  higher 
powers  can  therefore  almost  always  be  neglected  even  in  very 
long  transmission  lines,  and  the  equation  (9)  approximated  to: 


ZoFo,  Zo6F,          do) 

-f-  ~~2 —  |  ~^~  ^o^o  |  1  4~  ' — ^ —  f 


12 


178         ALTERNATING-CURRENT  PHENOMENA 

These  equations  are  simpler  than  those  often  given  by  repre- 
senting the  line  capacity  by  a  condenser  shunted  across  the  middle 
of  the  line,  and  are  far  more  exact.  They  give  the  generator 
voltage  and  current,  EI  repectively  /i,  by  the  step-down  voltage 

and  current,  EQ  and  IQ  respectively. 

Inversely,  if  EQ  and  70  are  chosen  as  the  values  at  the  generator 

end,  the  values  at  the  step-down  end  are  given  by  substituting 
Z  =  —  IQ  in  equations  (8),  as: 


.  v  I  i  _.    ZVY<>\ 

\   —  £SQ  }   1  -j  --  ~  - 


i 

T 


1  a 


1    _1_      °(]  V  J?        1 

I   H  --    -        —    JL  0^0  i    1 


(11) 


2    j       »ru  i         6 

Neglecting  the  line  conductance :  go  =  '0,  gives  : 

and:  ZQ  =  7*0  -f-  JXQ 

hence,  substituted  in  equations  (10)  and  (11),  and  expanded,  gives 

oZ0         JVol 


hr   }  .  ,  „          ^  ,       (12) 

.  Oofo  I 


where  the  upper  sign  holds,  if  E0)  IQ  are  at  the  step-down  end, 
EI,  Ii  at  the  generator  end  of  the  line,  and  the  lower  sign  holds, 
if  EO,  IQ  are  at  the  generator  end,  EI,  I\  at  the  step-down  end  of 

the  line. 

As  seen,  the  equations  (12)  are  just  as  simple  as  those  of  a 
circuit  containing  the  resistance,  inductance  and  capacity  lo- 
calized, and  are  amply  exact  for  practically  all  cases.  Where  a 
still  closer  approximation  should  be  required,  the  next  term  of 

equations  (8)  and  (9)  may  be  included. 

7  V 
In  many  cases,  the  —  ^  term  in   (10)  and  (11)  may  also  be 

dropped,  giving  the  still  simpler  equation: 


/-|0\ 

V  \  ^    ' 

T      1  _L_  °1  -J-   V   77 

=   IQ    1  H  --  :r—  }   ±    YQ£JQ 


CHAPTER  XVI 

POWER,  AND  DOUBLE-FREQUENCY  QUANTITIES  IN 

GENERAL 

135.  Graphically,  alternating  currents  and  voltages  are  repre- 
sented by  vectors,  of  which  the  length  represents  the  intensity, 
the  direction  the  phase  of  the  alternating  wave.  The  vectors 
generally  issue  from  the  center  of  coordinates. 

In  the  topographical  method,  however,  which  is  more  con- 
venient for  complex  networks,  as  interlinked  polyphase  circuits, 
the  alternating  wave  is  represented  by  the  straight  line  between 
two  points,  these  points  representing  the  absolute  values  of 
potential  (with  regard  to  any  reference  point  chosen  as  coordi- 
nate center),  and  their  connection  the  difference  of  potential  in 
phase  and  intensity. 

Algebraically  these  vectors  are  represented  by  complex  quan- 
tities. The  impedance,  admittance,  etc.,  of  the  circuit  is  a  com- 
plex quantity  also,  in  symbolic  denotation. 

Thus  current,  voltage,  impedance,  and  admittance  are  related 
by  multiplication  and  division  of  complex  quantities  in  the  same 
way  as  current,  voltage,  resistance,  and  conductance  are  related 
by  Ohm's  law  in  direct-current  circuits. 

In  direct-current  circuits,  power  is  the  product  of  current  into 
voltage.  In  alternating-current  circuits,  if 

E  =  e*+je", 
I  =  {i  +  ^11, 
the  product, 

P0  =  El 


is  not  the  power;  that  is,  multiplication  and  division,  which  are 
correct  in  the  inter-relation  of  current,  voltage,  impedance,  do 
not  give  a  correct  result  in  the  inter-relation  of  voltage,  current, 
power.  The  reason  is,  that  E  and  I  are  vectors  of  the  same  fre- 
quency, and  Z  a  constant  numerical  factor  or  "  operator,"  which 
thus  does  not  change  the  frequency. 

179 


180         ALTERNATING-CURRENT  PHENOMENA 

The  power,  P,  however,  is  of  double  frequency  compared  with 
E  and  /,  that  is,  makes  a  complete  wave  for  every  half  wave  of 
E  or  7,  and  thus  cannot  be  represented  by  a  vector  in  the  same 
diagram  with  E  and  I. 

PQ  =  El  is  a,  quantity  of  the  same  frequency  with  E  and  /,  and 

thus  cannot  represent  the  power. 

136.  Since  the  power  is  a  quantity  of  double  frequency  of  E 
and  7,  and  thus  a  phase  angle,  6,  in  E  and  7  corresponds  to  a 
phase  angle,  2  6,  in  the  power,  it  is  of  interest  to  investigate  the 
product,  El,  formed  by  doubling  the  phase  angle. 

Algebraically  it  is, 

p  =  El  =  (el  +  jen)(il  +  jin) 


Since  j2  =  -  1,  that  is,  180°  rotation  for  E  and  7,  for  the  double- 
frequency  vector,  P,  f  =  +1,  or  360°  rotation,  and 

j  X  1  =  j, 
1  X  j  =  -  j. 

That  is,  multiplication  with  j  reverses  the  sign,  since  it  denotes 
a  rotation  by  180°  for  the  power,  corresponding  to  a  rotation  of 
90°  for  E  and  7. 
Hence,  substituting  these  values,  we  have 

p  =  [El]  =  (elil  +  ellin)  +  j(ellil  -  eli11). 

The  symbol  [El]  here  denotes  the  transfer  from  the  frequency 
of  E  and  7  to  the  double  frequency  of  P. 

The  product,  P  =  [El],  consists  of  two  components:  the  real 
component, 

pi  =  [El]1  =  (gH'i  +  6"t»); 

and  the  imaginary  component, 

JP*  =  j[EI\i  =  JO11*1  -  eli"). 
The  component, 

P1  =  [El]1  =  (eW  +  enin), 

is  the  true  or  "effective"  power  of  the  circuit,  =  El  cos  (#7). 
The  component, 

pi  =  [EI]i  =  (ellil  -  eH'11), 

is  what  may  be  called  the  "reactive  power,"  or  the  wattless  or 
quadrature  volt-amperes  of  the  circuit,  =  El  sin  (El). 


DOUBLE-FREQUENCY  QUANTITIES  181 

The  real  component  will  be  distinguished  by  the  index  1;  the 
imaginary  or  reactive  component  by  the  index,  j. 

By  introducing  this  symbolism,  the  power  of  an  alternating 
circuit  can  be  represented  in  the  same  way  as  in  the  direct-cur- 
rent circuit,  as  the  symbolic  product  of  current  and  voltage. 

Just  as  the  symbolic  expression  of  current  and  voltage  as  com- 
plex quantity  does  not  only  give  the  mere  intensity,  but  also  the 
phase, 

E  =  el  +e11 


T-»  /      2  2 

E  =  -^i   +  eii 

gll 
tan  0  =  — f , 

so  the  double-frequency  vector  product  P  =  [E7]  denotes  more 
than  the  mere  power,  by  giving  with  its  two  components,  P1  = 
[El]1  and  Py  =  [El]1',  the  true  power  volt-ampere,  or  ''effective 
power,"  and  the  wattless  volt-amperes,  or  "reactive  power." 
If 

E  =  e1  +  jen, 
/  =  ;i  +  #", 
then 


e1    -f-  eLL  > 

r~*     ^ 

and 

pi  =  [^/]  i  =  (gH-i  +  euin), 

py  =  [i»7]/  =  (e^i1  -  elin), 
or 


where  Pa  =  total  volt-amperes  of  circuit.     That  is, 

The  effective  power,  P1,  and  the  reactive  power,  Pj,  are  the  two 

rectangular  components  of  the  total  apparent  power,  Pa,  of  the 

circuit. 

Consequently, 

In  symbolic  representation  as  double-frequency  vector  products, 

powers   can   be   combined  and  resolved  by  the   parallelogram   of 

vectors  just  as   currents   and  voltages  in   graphical  or  symbolic 

representation. 


182         ALTERNATING-CURRENT  PHENOMENA 

The  graphical  methods  of  treatment  of  alternating-current 
phenomena  are  here  extended  to  include  double-frequency 
quantities,  as  power,  torque,  etc. 

P1 

p  =  p-  =  cos  0  =  power-factor. 

•*   a 

pi 

q  —  p-  =  sin  0  =  induction  factor 
#  • 

of  the  circuit,  and  the  general  expression  of  power  is 
P  =  Pa  (p  +  jg)  =  Pa  (cos  0  +  j  sin  0). 

137.  The  introduction  of  the  double-frequency  vector  product, 
P  =  [El],  brings  us  outside  of  the  limits  of  algebra,  however, 
and  the  commutative  principle  of  algebra,  a  X  b  =  b  X  a, 
does  not  apply  any  more,  but  we  have 

[El]  unlike  [IE] 
since 

[El]  =  [EIY+j[EIV 

[IE]  =  [IEV  +  j[IEV 
we  have 

[El]1  =  [IE]1 


that  is,  the  imaginary  component  reverses  its  sign  by  the  inter- 
change of  factors. 

The  physical  meaning  is,  that  if  the  reactive  power,  [El]1', 
is  lagging  with  regard  to  E,  it  is  leading  with  regard  to  /. 

The  reactive  component  of  power  is  absent,  or  the  total 
apparent  power  is  effective  power,  if 

[El]*'  =  (ellil  -  elin)  =  0; 

that  is, 

e^_       i^ 
el   :=  il 
or, 

tan  (E)  =  tan  (/>; 

that  is,  E  and  /  are  in  phase  or  in  opposition. 

The  effective  power  is  absent,  or  the  total  apparent  power 
reactive,  if 

[El]1  =  (elil  +  elHn)  =  0; 


DOUBLE-FREQUENCY  QUANTITIES  183 

that  is,  e^_       _  V_ 

e1  "         in 
or, 

tan  E  =  —  cot  /; 

that  is,  E  and  /  are  in  quadrature. 

The  reactive  component  of  power  is  lagging  (with  regard  to 
E  or  leading  with  regard  to  /)  if 


0, 
and  leading  if 

[EIV<  0. 

The  effective  power  is  negative,  that  is,  power  returns,  if 

[EI]l<  0. 
We  have, 

IE,-I]  =  [-E,i}  =  -[Ei] 
I-  E,  -  i]  =  +  [Ei]  ' 

that  is,  when  representing  the  power  of  a  circuit  or  a  part  of  a 
circuit,  current  and  voltage  must  be  considered  in  their  proper 
relative   phases,   but   their   phase  relation   with   the  remaining 
part  of  the  circuit  is  immaterial. 
We  have  further, 

[E,  in  =  -  j  [E,  I]  =  [E,  IV  -  j  [E,  IV 


[JE,JI]  =  [E}I] 

138.  Expressing  voltage  and  current  in  polar  coordinates; 

E  =  el+  je11  =  e  (cos  a  +  j  sin  a) 
I  =  p  -j-  ji"  =  i  (cos  |8  +  j  sin  0) 

gives  the  vector  power: 

P  =  ei{  (cos  a  cos  0  +  j2  sin  a  sin  0)  -f-  (j  sin  a  cos  /?  +  cos  aj  sin 

and  since,  by  the  change  to  double  frequency: 

+  j2  =  +  1 

+  aj  =  -  ja 
it  is: 
P  =  ei  {  (cos  a  cos  0  +  sin  a  sin  0)  +  jXsin  a  sin  0  —  cos  a  cos 

P  =  ei  {cos  («  -  0)  +  j  sin  (a  -  0)} 


184         ALTERNATING-CURRENT  PHENOMENA 

and: 

the  effective  power: 

P1  =  ei  cos  (a  —  /3) 
the  reactive  power: 

P>  =  ei  sin  (a  —  0) 

We  thus  must  note  the  distinction: 

E  =  ZI  =  (r  +  jx)  (i1  +  ji11)  =  zi  (cos  7  +  j  sin  7)  (cos  0  +  j  sin  £) 
=  (n1'-  xi11)  +  j  (n11  +  xil)  =  w  {cos  (7  +  0)  +  j  sin  (7  +  0) } 
and: 

P  =  [£,/]  =  [£,/P+j[£,/]' 

=  [(e1  +  je11),  C*1  +  ft11}}  =  ei  [(cos  a  +  j  sin  a),  (cos  0  +j  sin  /5)] 
=  (e1*1  +  e11*11)  +  j  (ellil  -  eli11)  =  cz  {cos  («-£)  + 

j  sin  (a  -  j8)  J 
139.  If  P!  =  [J^i/J,  P2  =  [£-2/2]   .    .    .   Pn  =  [Enln] 

are  the  symbolic  expressions  of  the  power  of  the  different  parts 
of  a  circuit  or  network  of  circuits,  the  total  power  of  the  whole 
circuit  or  network  of  circuits  is 

P  =  Pi   +  P2   +....+  Pn, 

pi    =X+P12+      ....      +Pn1, 

Pi  =  P2y  _f-  p8y  .    .    .    .    +  pn/. 

In  other  words,  the  total  power  in  symbolic  expression  (effect- 
ive as  well  as  reactive)  of  a  circuit  or  system  is  the  sum  of  the 
powers  of  its  individual  components  in  symbolic  expression. 

The  first  equation  is  obviously  directly  a  result  from  the  law 
of  conservation  of  energy. 

One  result  derived  herefrom  is,  for  instance: 

If  in  a  generator  supplying  power  to  a  system  the  current  is 
out  of  phase  with  the  e.m.f.  so  as  to  give  the  reactive  power 
P*,  the  current  can  be  brought  into  phase  with  the  generator 
e.m.f.  or  the  load  on  the  generator  made  non-inductive  by  in- 
serting anywhere  in  the  circuit  an  apparatus  producing  the  react- 
ive power — P1;  that  is,  compensation  for  wattless  currents  in  a 
system  takes  place  regardless  of  the  location  of  the  compensating 
device. 

Obviously,  wattless  currents  exist  between  the  compensating 
device  and  the  source  of  wattless  currents  to  be  compensated 
for,  and  for  this  reason  it  may  be  advisable  to  bring  the  com- 
pensator as  near  as  possible  to  the  circuit  to  be  compensated. 


DOUBLE-FREQUENCY  QUANTITIES  185 

140.  Like  power,  torque  in  alternating  apparatus  is  a  double- 
frequency  vector  product  also,  of  magnetism  and  m.m.f.  or 
current,  and  thus  can  be  treated  in  the  same  way. 

In  an  induction  motor,  for  instance,  the  torque  is  the  product 
of  the  magnetic  flux  in  one  direction  into  the  component  of 
secondary  current  in  phase  with  the  magnetic  flux  in  time,  but 
in  quadrature  position  therewith  in  space,  times  the  number  of 
turns  of  this  current,  or  since  the  generated  e.m.f.  is  in  quad- 
rature and  proportional  to  the  magnetic  flux  and  the  number 
of  turns,  the  torque  of  the  induction  motor  is  the  product  of 
the  generated  e.m.f.  into  the  component  of  secondary  current 
in  quadrature  therewith  in  time  and  space,  or  the  product  of 
the  secondary  current  into  the  component  of  generated  e.m.f. 
in  quadrature  therewith  in  time  and  space. 

Thus,  if 

El  =  el  +  je11  =  generated  e.m.f.  in  one  direction  in  space, 

1  2  =  il  +  jin  =  secondary  current  in  the  quadrature  direction 


in  space, 
the  torque  is 


D  = 


By  this  equation  the  torque  is  given  in  watts,  the  meaning 
being  that  D  =  [EI]>  is  the  power  which  would  be  exerted  by 
the  torque  at  synchronous  speed,  or  the  torque  in  synchronous 
watts. 

The  torque  proper  is  then 


vy  fl  £*T*A 

p  =  number  of  pairs  of  poles  of  the  motor. 
/  =  frequency. 

In  the  polyphase  induction  motor,   if  72  =  il  +  jin  is  the 
secondary  current  in  quadrature  position,  in  space,  to  e.m.f.  E\, 
the  current  in  the  same  direction  in  space  as  E\  is  7i  =  jlz  — 
—  iu  -f-  ji1-  thus  the  torque  can  also  be  expressed  as 
D  =  [EJi]1  =  ellil  -  cH'11. 

It  is  interesting  to  note  that  the  expression  of  torque, 

D  =  [Eiy, 

and  the  expression  of  power, 

P  =  (EIV, 


186         ALTERNATING-CURRENT  PHENOMENA 

are  the  same  in  character,  but  the  former  is  the  imaginary,  the 
latter  the  real  component.  Mathematically,  torque,  in  syn- 
chronous watts,  can  so  be  considered  as  imaginary  power,  and 
inversely.  Physically,  " imaginary"  means  quadrature  compo- 
nent; torque  is  defined  as  force  times  leverage,  that  is,  force 
times  length  in  quadrature  position  with  force;  while  energy  is 
defined  as  force  times  length  in  the  direction  of  the  force.  Ex- 
pressing quadrature  position  by  "imaginary,"  thus  gives  torque 
of  the  dimension  of  imaginary  energy;  and  "synchronous  watts," 
which  is  torque  times  frequency,  or  torque  divided  by  time,  thus 
becomes  of  the  dimension  of  imaginary  power.  Thus,  in  its 
complex  imaginary  form,  the  vector  product  of  force  and  length 
contains  two  quadrature  components,  of  which  the  one  is  energy, 
the  other  is  torque: 

P  =  (f,l]  =  [f,lV+Af,l}> 
and 

[/,  I}1  =  energy 
[/,  I]*  =  torque. 


SECTION  IV 
INDUCTION  APPARATUS 


CHAPTER  XVII 
THE   ALTERNATING-CURRENT   TRANSFORMER 

141.  The  simplest  alternating-current  apparatus  is  the  trans- 
former.    It  consists  of  a  magnetic  circuit  interlinked  with  two 
electric  circuits,  a  primary  and  a  secondary.     The  primary  circuit 
is  excited  by  an  impressed  e.m.f.,  while  in  the  secondary  circuit 
an  e.m.f.  is  generated.     Thus,  in  the  primary  circuit  power  is 
consumed,   and  in  the  secondary  a  corresponding  amount  of 
power  is  produced. 

Since  the  same  magnetic  circuit  is  interlinked  with  both 
electric  circuits,  the  e.m.f.  generated  per  turn  must  be  the  same 
in  the  secondary  as  in  the  primary  circuit;  hence,  the  primary 
generated  e.m.f.  being  approximately  equal  to  the  impressed 
e.m.f.,  the  e.m.fs.  at  primary  and  at  secondary  terminals  have 
approximately  the  ratio  of  their  respective  turns.  Since  the 
power  produced  in  the  secondary  is  approximately  the  same 
as  that  consumed  in  the  primary,  the  primary  and  secondary 
currents  are  approximately  in  inverse  ratio  to  the  turns. 

142.  Besides  the  magnetic  flux  interlinked  with  both  electric 
circuits — which  flux,  in  a  closed  magnetic  circuit  transformer, 
has  a  circuit  of  low  reluctance — a  magnetic  cross-flux  passes 
between  the  primary  and  secondary  coils,  surrounding  one  coil 
only,  without  being  interlinked  with  the  other.     This  magnetic 
cross-flux  is  proportional  to  the  current  in  the  electric  circuit, 
or  rather,  the  ampere-turns  or  m.m.f.,  and  so  increases  with  the 
increasing  load   on   the  transformer,   and   constitutes   what  is 
called   the   self-inductive   or   leakage   reactance   of   the   trans- 
former;  while   the   flux   surrounding   both   coils   may   be   con- 
sidered   as    mutual    inductive    reactance.     This    cross-flux    of 
self-induction  does  not  generate  e.m.f.  in  the  secondary  circuit, 

187 


188         ALTERNATING-CURRENT  PHENOMENA 

and  is  thus,  in  general,  objectionable,  by  causing  a  drop  of 
voltage  and  a  decrease  of  output.  It  is  this  cross-flux,  how- 
ever, or  flux  of  self-inductive  reactance,  which  is  utilized  in 
special  transformers,  to  secure  automatic  regulation,  for  con- 
stant power,  or  for  constant  current,  and  in  this  case  is  exagger- 
ated by  separating  primary  and  secondary  coils.  In  the  con- 
stant potential  transformer,  however,  the  primary  and  secondary 
coils  are  brought  as  near  together  as  possible,  or  even  inter- 
spersed, to  reduce  the  cross-flux. 

There  is,  however,  a  limit,  to  which  it  is  safe  to  reduce  the 
cross-flux,  as  at  short-circuit  at  the  secondary  terminals,  it  is  the 
e.m.f.  of  self-induction  of  this  cross-flux  which  limits  the  current, 
and  with  very  low  self-induction,  these  currents  may  become 
destructive  by  their  mechanical  forces.  Therefore  experience 
shows  that  in  large  power  transformers  it  is  not  safe  to  go  below 
4  to  6  per  cent,  cross-flux. 

As  will  be  seen,  by  the  self-inductive  reactance  of  a  circuit,  not 
the  total  flux  produced  by,  and  interlinked  with,  the  circuit  is 
understood,  but  only  that  (usually  small)  part  of  the  flux  which 
surrounds  one  circuit  without  interlinking  with  the  other  circuit. 

143.  The  alternating  magnetic  flux  of  the  magnetic  circuit 
surrounding  both  electric  circuits  is  produced  by  the  combined 
magnetizing  action  of  the  primary  and  of  the  secondary  current. 

This  magnetic  flux  is  determined  by  the  e.m.f.  of  the  trans- 
former, by  the  number  of  turns,  and  by  the  frequency. 

If 

$  =  maximum  magnetic  flux, 

/  =  frequency, 

n  =  number  of  turns  of  the  coil, 

the  e.m.f.  generated  in  this  coil  is 


E  =  V2Vn$  10~8  =  4.44  fn&  lO"8  volts; 

hence,  if  the  e.m.f.,  frequency,  and  number  of  turns  are  de- 
termined, the  maximum  magnetic  flux  is 

ff  108 
"  V2*/n" 

To  produce  the  magnetism,  <f>,  of  the  transformer,  a  m.m.f. 
of  F  ampere-turns  is  required,  which  is  determined  by  the  shape 
and  the  magnetic  characteristic  of  the  iron,  in  the  manner  dis- 
cussed in  Chapter  XII. 


ALTERNATING-CURRENT  TRANSFORMER       189 

144.  Consider  as  instance,  a  closed  magnetic  circuit  transformer. 

<j> 
The  maximum  magnetic   induction  is  B  =  -j,  where  A  =  the 

cross-section  of  magnetic  circuit. 

To  induce  a  magnetic  density,  J5,  a  magnetizing  force  of  / 

ampere-turns  maximum  is  required,  or  —=  ampere-turns  effect- 


ive, per  unit  length  of  the  magnetic  circuit;  hence,  for  the  total 
magnetic  circuit,  of  length,  I, 

v 


or 


— -7-  ampere-turns; 

v  2 

F          If 
I  =  -  = y=  amp.  eff. 

n       n-v/2 


where  n  =  number  of  turns. 

At  no-load,  or  open  secondary  circuit,  this  m.m.f.,  F,  is  fur- 
nished by  the  exciting  current,  70o,  improperly  called  the  leakage 
current ,  of  the  transformer;  that  is,  that  small  amount  of  primary 
current  which  passes  through  the  transformer  at  open  secondary 
circuit. 

In  a  transformer  with  open  magnetic  circuit,  such  as  the 
"hedgehog"  transformer,  the  m.m.f.,  F,  is  the  sum  of  the  m.m.f. 
consumed  in  the  iron  and  in  the  air  part  of  the  magnetic  circuit 
(see  Chapter  XII). 

The  power  component  of  the  exciting  current  represents  the 
power  consumed  by  hysteresis  and  eddy  currents  and  the  small 
ohmic  loss. 

The  exciting  current  is  not  a  sine  wave,  but  is,  at  least  in 
the  closed  magnetic  circuit  transformer,  greatly  distorted  by 
hysteresis,  though  less  so  in  the  open  magnetic  circuit  trans- 
former. It  can,  however,  be  represented  by  an  equivalent  sine 
wave,  7oo,  of  equal  intensity  and  equal  power  with  the  distorted 
wave,  and  a  wattless  higher  harmonic,  mainly  of  triple  frequency. 

Since  the  higher  harmonic  'is  small  compared  with  the  total 
exciting  current,  and  the  exciting  current  is  only  a  small  part 
of  the  total  primary  current,  the  higher  harmonic  can,  for  most 
practical  cases,  be  neglected,  and  the  exciting  current  repre- 
sented by  the  equivalent  sine  wave. 

This  equivalent  sine  wave,  70o,  leads  the  wave  of  magnetism, 
<$>,  by  an  angle,  a,  the  angle  of  hysteretic  advance  of  phase,  and 


190         ALTERNATING-CURRENT  PHENOMENA 

consists  of  two  components — the  hysteretic  power  current 
in  quadrature  with  the  magnetic  flux,  and  therefore  in  phase 
with  the  generated  e.m.f.  =  700  sin  a]  and  the  magnetizing 
current,  in  phase  with  the  magnetic  flux,  and  therefore  in  quad- 
rature with  the  generated  e.m.f.,  and  so  wattless,  =  70o  cos  a. 

The  exciting  current,  70o,  is  determined  from  the  shape  and 
magnetic  characteristic  of  the  iron,  and  the  number  of  turns; 
the  hysteretic  power  current  is 

power  consumed  in  the  iron 

/on  Sin  a  = — ; » 

generated  e.m.f. 

145.  Graphically,  the  polar  diagram  of  m.m.fs.,  of  a  trans- 
former is  constructed  thus : 

Let,  in  Fig.  102,  0$  =  the  magnetic  flux  in  intensity  and 
phase  (for  convenience,  as  intensities,  the  effective  values  are 


FIG.  102. 

used  throughout),  assuming  its  phase  as  the  downwards  vertical; 
that  is,  counting  the  time  from  the  moment  where  the  rising 
magnetism  passes  its  zero  value. 

Then  the  resultant  m.m.f.  is  represented  by  the  vector,  OF, 
leading  0$  by  the  angle,  FO&  =  a. 

The  generated  e.m.fs.  have  the  phase  180°,  that  is,  are  plotted 
toward  the  left,  and  represented  by  the  vectors,  OE'Q  and  OE\. 

If,  now,  0'  =  angle  of  lag  in  the  secondary  circuit,  due  to  the 
total  (internal  and  external)  secondary  reactance,  the  secondary 
current,  7i,  and  hence  the  secondary  m.m.f.,  F\  =  tti  7i  lag 
behind  E'i  by  an  angle  0',  and  have  the  phase,  180°  +  0',  repre- 


ALTERNATING-CURRENT  TRANSFORMER       191 

sented  by  the  vector  OFlf  Constructing  a  parallelogram  of 
m.m.fs.,  with  OF  as  the  diagonal  and  OFi  as  one  side,  the  other 
side  or  OF0  is  the  primary  m.m.f.,  in  intensity  and  phase,  and 
hence,  dividing  by  the  number  of  primary  turns,  n0>  the  primary 

...     f          ^o 

current  is  io  =  — • 
ft  4 

To  complete  the  diagram  of  e.m.fs.,  we  have  now, 

In  the  primary  circuit: 

e.m.f.  consumed  by  resistance  is  I0r0,  in  phase  with  /0,  and 
represented  by  the  vector,  OEro; 

e.m.f.  consumed  by  reactance  is  IOXQ,  90°  ahead  of  70,  and 
represented  by  the  vector,  OE XQ', 

e.m.f.  consumed  by  induced  e.m.f.  is  E',  equal  and  opposite 
to  E'o,  and  represented  by  the  vector,  OE' . 

Hence,jthe  total  primary  impressed  e.m.f.  'by  combination  of 
OEro,  OEXQ}  and  OE'  by  means  of  the  parallelogram  of  e.m.fs.  is 


EQ  =  OE0, 

and   the   difference   of  phase   between  the   primary  impressed 
e.m.f.  and  the  primary  current  is 


In  the  secondary  circuit: 

Counter  e.m.f.  of  resistance  is  I\r\  in  opposition  with  /i,  and 
represented  by  the  vector,  OE'ri', 

Counter  e.m.f.  of  reactance  is  IiXi,  90°  behind  /i,  and  repre- 
sented by  the  vector,  OE'X1. 

Generated  e.m.fs.,  E\,  represented  by  the  vector,  OE'V 

Hence,  the  secondary  terminal  voltage,  by  combination  of 
OE'ri,  OE'X1  and  OE'i  by  means  of  the  parallelogram  of  e.m.fs. 
is 

E,  =  OEi, 

and  the  difference  of  phase  between  the  secondary  terminal 
voltage  and  the  secondary  current  is 


As  seen,  in  the  primary  circuit  the  "components  of  impressed 
e.m.f.  required  to  overcome  the  counter  e.m.fs."  were  used  for 
convenience,  and  in  the  secondary  circuit  the  "  counter  e.m.fs." 


192 


ALTERNATING-CURRENT  PHENOMENA 


146.  In  the  construction  of  the  transformer  diagram,  it  is 
usually  preferable  not  to  plot  the  secondary  quantities,  current 
and  e.m.f.,  direct,  but  to-  reduce  them  to  correspondence  with 

the  primary  circuit  by  multiplying  by  the  ratio  of  turns,  a  =  — , 

for  the  reason  that  frequently  primary  and  secondary  e.m.fs., 
etc.,  are  of  such  different  magnitude  as  not  to  be  easily  repre- 
sented on  the  same  scale;  or  the  primary  circuit  may  be  reduced 
to  the  secondary  in  the  same  way.  In  either  case,  the  vectors 
representing  the  two  generated  e.m.fs.  coincide,  or  OE'i  =  OE'Q. 


FIG.  103. 

Figs.  103  to  109  give  the  polar  diagram  of  a  transformer  having 
the  constants,  reduced  to  the  secondary  circuit: 

r0  =  0.2  ohm,  60  =  0.173  mhos, 

x0  =  0.33  ohm,  E\  =  100  volts, 

7*1  =  0.167  ohm,  /i  =  60  amp., 
xi  =  0.25  ohm,  a  =  30°. 

<7o  =  0.100  mhos, 

For  the  conditions  of  secondary  circuit: 

8'i  =  80°  lag        in  Fig.  103  6\  =  20°  lead  in  Fig.  107 

50°  lag              "     104  50°  lead       "      108 

20°  lag               "     105  80°  lead       "      109 
0,  or  in  phase,  ' '     106 

As  shown,  with  a  change  of  0'i  the  other  quantities,  EQ,  /i, 
Jo,  etc.,  change  in  intensity  and  direction.     The  loci  described 


ALTERNATING-CURRENT  TRANSFORMER       193 


Ek. 


FIG.  104. 


FIG.  105. 


13 


FIG.  106. 


194         ALTERNATING-CURRENT  PHENOMENA 


FIG.  107. 


FIG.  108. 


ALTERNATING-CURRENT  TRANSFORMER       195 


FIG.  111. 


196         ALTERNATING-CURRENT  PHENOMENA 

by  them  are  circles,  and  are  shown  in  Fig.  110,  with  the  point 
corresponding  to  non-inductive  load  marked.  The  part  of  the 
locus  corresponding  to  a  lagging  secondary  current  is  shown 
in  thick  full  lines,  and  the  part  corresponding  to  leading  current 
in  thin  full  lines. 

147.  This  diagram  represents  the  condition  of  constant 
secondary  generated  e.m.f.,  E'i,  that  is,  corresponding  to  a  con- 
stant maximum  magnetic  flux. 

By  changing  all  the  quantities  proportionally  from  the  dia- 
gram of  Fig.  110,  the  diagrams  for  the  constant  primary  im- 


FIG.  113. 

pressed  e.m.f.  (Fig.  Ill),  and  for  constant  secondary  terminal 
voltage  (Fig.  112),  are  derived.  In  these  cases,  the  locus  gives 
curves  of  higher  order. 

Fig.  113  gives  the  locus  of  the  various  quantities  when  the 
load  is  changed  from  full-load,  /i  =  60  amp.  in  a  non-inductive 
secondary  external  circuit,  to  no-load  or  open-circuit: 

(a)  By  increase  of  secondary  current;  (6)  by  increase  of 
secondary  inductive  resistance;  (c)  by  increase  of  secondary 
condensive  reactance. 

As  shown  in  (a),  the  locus  of  the  secondary  terminal  voltage, 
Eit  and  thus  of  EQ,  etc.,  are  straight  lines;  and  in  (6)  and  (c), 
parts  of  one  and  the  same  circle;  (a)  is  shown  in  full  lines,  (6)  in 
heavy  full  lines,  and  (c)  in  light  full  lines.  This  diagram  corre- 
sponds to  constant  maximum  magnetic  flux;  that  is,  to  constant 
secondary  generated  e.m.f.  The  diagrams  representing  constant 


ALTERNATING-CURRENT  TRANSFORMER       197 

primary  impressed  e.m.f.  and  constant  secondary  terminal 
voltage  can  be  derived  from  the  above  by  proportionality. 

148.  It  must  be  understood,  however,  that  for  the  purpose 
of  making  the  diagrams  plainer,  by  bringing  the  different  values 
to  somewhat  nearer  the  same  magnitude,  the  constants  chosen 
for  these  diagrams  represent  not  the  magnitudes  found  in  actual 
transformers,  but  refer  to  greatly  exaggerated  internal  losses. 

In  practice,  about  the  following  magnitudes  would  be  found: 

r0  =  0.01        ohm;  xi  =  0.00025  ohm; 

X0  =  0.033      ohm;  gQ  =  0.001      mho; 

ri  =  0.00008  ohm;  b0  =  0.00173  mho; 

that  is,  about  one-tenth  as  large  as  assumed.  Thus  the  changes 
of  the  values  of  E0,  E\,  etc.,  under  the  different  conditions 
will  be  very  much  smaller. 

Symbolic  Method 

149.  In  symbolic  representation   by  complex  quantities  the 
transformer  problem  appears  as  follows: 

The  exciting  current,  /oo,  of  the  transformer  depends  upon 
the  primary  e.m.f.,  which  dependence  can  be  represented  by  an 
admittance,  the  "primary  admittance,"  Fo  =  go  —  jb0,  of  the 
transformer. 

The  resistance  and  reactance  of  the  primary  and  the  secondary 
circuit  are  represented  in  the  impedance  by 

ZQ  =  r0  +  jx0,     and     Zi  =  n  +  jxi. 

Within  the  limited  range  of  variation  of  the  magnetic  density 
in  a  constant-potential  transformer,  admittance  and  impedance 
can  usually,  and  with  sufficient  exactness,  be  considered  as 
constant. 

Let 

n o  =  number  of  primary  turns  in  series; 
HI  =  number  of  secondary  turns  in  series; 

nQ  . 

a  =  —  =  ratio  of  turns; 
Hi 

YQ  =  0o  —  jbo  =  primary  admittance 

Exciting  current 
~  Primary  induced  e.m.f. ' 


198         ALTERNATING-CURRENT  PHENOMENA 

ZQ  =  TO  +  jxo  =  primary  impedance 

_  e.m.f.  consumed  in  primary  coil  by  resistance  and  reactance  . 
Primary  current 

Zi  =  r\  +  jxi  =  secondary  impedance 

_  e.m.f.  consumed  in  secondary  coil  by  resistance  and  reactance  m 
Secondary  current 

where  the  reactances,  XQ  and  Xi,  refer  to  the  true  self-induction 
only,  or  to  the  cross-flux  passing  between  primary  and  second- 
ary coils;  that  is,  interlinked  with  one  coil  only. 
Let  also 

Y  =  g  —jb  =  total    admittance    of    secondary    circuit,    in- 

cluding the  internal  impedance; 
EQ  =  primary  impressed  e.m.f.  ; 
E'  =  e.m.f.  consumed  by  primary  counter  e.m.f.; 
Ei  =  secondary  terminal  voltage; 
E'i  =  secondary  generated  e.m.f.; 
IQ  =  primary  current,  total; 
I  oo  =  primary  exciting  current; 
1  1  =  secondary  current. 

Since   the   primary   counter   e.m.f.,   EQ',  and  the  secondary 
generated  e.m.f.,  E'i,  are  proportional  by  the  ratio  of  turns,  a, 

E'0  =  +  aE\.  (1) 

E'0  =  -  E'. 

The  secondary  current  is 

Ii=  YE'  i.  (2) 

consisting  of  a  power  component,  gEi,  and  a  reactive  component, 

Wi, 

To   this   secondary   current    corresponds    the   component   of 
primary  current. 

:.     .        '       (3) 


.     . 

a  a 

The  primary  exciting  current  is 

/oo  =  YoE'.  (4) 


ALTERNATING-CURRENT  TRANSFORMER       199 
Hence,  the  total  primary  current  is 

/o  =  f'  o  +  /oo  (5) 

YE' 

=  -       + 


or> 

/0  =      -F+a*Fo  (6) 


The  e.m.f.  consumed  in  the  secondary  coil  by  the  internal 
impedance  is  Z-J\. 

The  e.m.f.  generated  in  the  secondary  coil  by  the  magnetic 
flux  is  E'i. 

Therefore,  the  secondary  terminal  voltage  is 

77T         77*'  7     T     * 

or,  substituting  (2),  we  have 

77?  TTF/       t  -t  *7      ~\7  \  f7\ 

Hi i  =  EJ  i{l  —  Zii  }•  (j) 

The  e.m.f.  consumed  in  the  primary  coil  by  the  internal  im- 
pedance is  ZQ/O. 

The  e.m.f.  consumed  in  the  primary  coil  by  the  counter  e.m.f. 

Therefore,  the  primary  impressed  e.m.f.  is 

EQ   =   E     -f-  Zo/0> 

or,  substituting  (6), 

EQ=     E'  ,         _      „     . 

(8) 


+  ZoFo 
150.  We  thus  have, 


f  } 

primary  e.m.f.,       EQ  =  -aE\  1 1+  Z0F0  +  "^~ } '  (8) 

secondary  e.m.f.,    El  =  E\  {1  -  ZrFj,  (7) 

E' 
primary  current,     I0  = {F  -f  a270},  (6) 


200         ALTERNATING-CURRENT  PHENOMENA 
secondary  current,  7i  =  YEi1,  (2) 

as  functions  of  the  secondary  generated  e.m.f.,  EI,  as  parameter. 


From  the  above  we  derive 

Ratio  of  transformation  of  e.m.fs.: 


Z0Y 


=  —  a 


(9) 


Ratio  of  transformations  of  currents : 


From  this  we  get,  at  constant  primary  impressed  e.m.f., 

EQ  —  constant; 
secondary  generated  e.m.f., 

1 


e.m.f.  generated  per  turn, 


dE    =  -- 

UQ 


secondary  terminal  voltage, 

EQ  1   -  ZiY 


primary  current, 


1  +  Z0Fo 


ZnF 


=  UQ 


1+Z0F0 


ZQY 


secondary  current, 


(ID 


At  constant  secondary  terminal  voltage, 
EI  =  const.; 


ALTERNATING-CURRENT  TRANSFORMER       201 

secondary  generated  e.m.f., 


1  - 


e.m.f.  generated  per  turn, 
primary  impressed  e.m.f., 

primary  current, 
secondary  current, 


m  1-ZiY' 


EQ=  - 


1  - 


Ei 


a     1  - 
Y 


(12) 


151.  Some  interesting  conclusions  can  be  drawn  from  these 
equations. 

The  apparent  impedance  of  the  total  transformer  is 


(13) 


+  z,. 


(14) 


Substituting  now,  —^  =  Y',  the  total  secondary  admittance, 


reduced  to  the  primary  circuit  by  the  ratio  of  turns,  it  is 


^  - 


YQ  +  Y' 

YQ  +  5"  is  the  total  admittance  of  a  divided  circuit  with  the 
exciting  current  of  admittance,  YQ,  and  the  secondary  current 
of  admittance,  Y'  (reduced  to  primary),  as  branches.  Thus, 

Yo—Y'  =  Z/0  (16) 


202         ALTERNATING-CURRENT  PHENOMENA 


is  the  impedance  of  this  divided  circuit,  and 

Zt  =  Z'0  +  Z0.  (17) 

That  is, 

The  alternate-current  transformer,  of  primary  admittance  YQ, 
total  secondary  admittance  Y,  and  primary  impedance  Z0,  is 
equivalent  to,  and  can  be  replaced  by,  a  divided  circuit  with  the 
branches  of  admittance  Y0,  the  exciting  current,  and  admittance 

Y 
Y'  =  -g,  the  secondary  current,  fed  over  mains  of  the  impedance 

ZQ,  the  internal  primary  impedance. 

This  is  shown  diagrammatically  in  Fig.  114. 


Generator 


Transformer 


FIG.  114. 

152.  Separating  now  the  internal  secondary  impedance  from 
the  external  secondary  impedance,  or  the  impedance  of  the 
consumer  circuit,  it  is 

1 


i  +  Z; 

where  Z  =  external  secondary  impedance, 


(18) 


Reduced  to  primary  circuit,  it  is 


,  =       =  a2Zi  +  a2Z 


That  is, 


(19) 


(20) 


ALTERNATING-CURRENT  TRANSFORMER       203 


An  alternate-current  transformer,  of  primary  admittance  Y0, 
primary  impedance  ZQ,  secondary  impedance  Z\,  and  ratio  of. 
turns  a}  can,  when  the  secondary  circuit  is  closed  by  an  impedance, 
Z  (the  impedance  of  the  receiver  circuit) ,  be  replaced,  and  is  equiva- 
lent to  a  circuit  of  impedance,  Z'  =  a2Z,  fed  over  mains  of  the 
impedance,  ZQ  +  Z'i,  where  Z\  =  a2Zi,  shunted  by  a  circuit  of 
admittance,  YQ,  which  latter  circuit  branches  off  at  the  points, 
a,  b,  between  the  impedances,  ZQ  and  Z\. 

This  is  represented  diagrammatically  in  Fig.  115. 


Generator 


i. 


Transformer    I  , 


FIG.  116. 

It  is  obvious,  therefore,  that  if  the  transformer  contains  sev- 
eral independent  secondary  circuits,  they  are  to  be  considered  as 
branched  off  at  the  points  a,  b,  in  diagram,  Fig.  115,  as  shown  in 
diagram,  Fig.  116. 

It  therefore  follows : 

An  alternate-current   transformer,  of  s  secondary   coils,  of  the 


204         ALTERNATING-CURRENT  PHENOMENA 


internal  impedances,  Z*,  Zi11, . . . Z^,  closed  by  external  secondary 
circuits  of  the  impedances,  Z1,  Z11, . . .  Zs,  is  equivalent  to  a  divided 
circuit  of  s  +  1  branches,  one  branch  of  admittance,  Y0,  the  excit- 
ing current,  the  other  branches  of  the  impedances,  Z\  +  Z7, 
Zi1  +  Z11, .  . .  Zi8  +  Z*j  the  latter  impedances  being  reduced  to 
the  primary  circuit  by  the  ratio  of  turns,  and  the  whole  divided 
circuit  being  Jed  by  the  primary  impressed  e.m.f.,  EQ,  over  mains 
of  the  impedance,  ZQ. 

Consequently,  transformation  of  a  circuit  merely  changes 
all  the  quantities  proportionally,  introduces  in  the  mains  the 
impedance,  ZQ  +  Z'i,  and  a  branch  circuit  between  ZQ  and  Z\, 
of  admittance  F0. 

Thus,  double  transformation  will  be  represented  by  diagram, 
Fig.  117. 

With  this  the  discussion  of  the  alternate-current  transformer 
ends,  by  becoming  identical  with  that  of  a  divided  circuit  con- 
taining resistances  and  reactances. 


Transformer 


Transformer 


Receiving 
Circuit 


FIG.  117. 

Such  circuits  have  been  discussed  in  detail  in  Chapter  IX, 
and  the  results  derived  there  are  now  directly  applicable  to  the 
transformer,  giving  the  variation  and  the  control  of  secondary 
terminal  voltage,  resonance  phenomena,  etc. 

Thus,  for  instance,  if  Z'i  =  ZQ,  and  the  transformer  contains 
an  additional  secondary  coil,  constantly  closed  by  a  condensive 
reactance  of  such  size  that  this  auxiliary  circuit,  together  with 
the  exciting  circuit,  gives  the  reactance,  —  XQ,  with  a  non-inductive 
secondary  circuit,  Z\  =  n,  we  get  the  condition  of  transformation 
from  constant  primary  potential  to  coristant  secondary  current, 
and  inversely. 


ALTERNATING-CURRENT  TRANSFORMER       205 

153.  As  seen,  the  alternating-current  transformer  is  charac- 
terized by  the  constants: 

Ratio  of  turns:  a  =  —  • 

f*i 

Exciting  admittance:  Y0  =  gQ  —  jb0. 

Self-inductive  impedances:  Z0  =  r0  +  jx0. 

Zi  =  ri  +  jxi. 

Since  the  effect  of  the  secondary  impedance  is  essentially  the 
same  as  that  of  the  primary  impedance  (the  only  difference 
being,  that  no  voltage  is  consumed  by  the  exciting  current  in  the 
secondary  impedance,  but  voltage  is  consumed  in  the  primary 
impedance,  though  very  small  in  a  constant-potential  trans- 
former), the  individual  values  of  the  two  impedances,  Z0  and  Zi, 
are  of  less  importance  than  the  resultant  or  total  impedance  of  the 
transformer,  that  is,  the  sum  of  the  primary  impedance  plus 
the  secondary  impedance  reduced  to  the  primary  circuit : 

Z'  =  Z0  +  a2Zi, 

and  the  transformer  accordingly  is  characterized  by  the  two 
constants : 

Exciting  admittance,  F0  =  go  —  j&o. 

Total  self-inductive  impedance,  Z'  =  rf  +  jx'. 

Especially  in  constant-potential  transformers  with  closed 
magnetic  circuit — as  usually  built — the  combination  of  both 
impedances  into  one,  Z',  is  permissible  as  well  within  the  errors 
of  observation. 

Experimentally,  the  exciting  admittance,  Fo  =  fifo  —  jbo,  and 
the  total  self-inductive  impedance,  Z'  =  r'  +  jx't  are  deter- 
mined by  operating  the  transformer  at  its  normal  frequency: 

1.  With    open    secondary    circuit,    and    measuring   volts   eo, 
amperes  io,  and  watts  WQ,  input — excitation  test. 

2.  With  the  secondary  short-circuited,  and  measuring  volts 
ei,  amperes  ii,  and  watts  pi,  input.     (In  this  case,  usually  a  far 
lower  impressed  voltage  is  required — impedance  test.) 

It  is  then: 


=  \A/o2  +  £fo 


r  = 


206         ALTERNATING-CURRENT  PHENOMENA 

If  a  separation  of  the  total  impedance  Zf  into  the  primary 
impedance  and  the  secondary  impedance  is  desired,  as  a  rule  the 
secondary  reactance  reduced  to  the  primary  can  be  assumed 
as  equal  to  the  primary  reactance  : 


except  if  from  the  construction  of  the  transformer  it  can  be  seen 
that  one  of  the  circuits  has  far  more  reactance  than  the  other, 
and  then  judgment  or  approximate  calculation  must  guide  in 
the  division  of  the  total  reactance  between  the  two  circuits. 

If  the  total  effective  resistance,  r',  as  derived  by  wattmeter, 
equals  the  sum  of  the  ohmic  resistances  of  primary  and  of 
secondary  reduced  to  the  primary: 

r'  =  r0  +  a2r, 

the  ohmic  resistances,  ro  and  n,  as  measured  by  Wheatstone 
bridge  or  by  direct  current,  are  used. 

If  the  effective  resistance  is  greater  than  the  resultant  of  the 
ohmic  resistances: 

r'  >  r0  +  aVi, 
the  difference: 

r"  =  r'  -  (r0  +  aVi) 

may  be  divided  between  the  two  circuits  in  proportion  to  the 
ohmic  resistances,  that  is,  the  effective  resistance  distributed 
between  the  two  circuits  in  the  proportion  of  their  ohmic  resist- 
ances, so  giving  the  effective  resistances  of  the  two  circuits, 
r'o  and  r'i,  by: 

r'o  -*-  r'i  =  r0  4-  n; 

or,  if  from  the  construction  of  the  transformer  as  the  use  of 
large  solid  conductors,  it  can  be  seen  that  the  one  circuit  is 
entirely  or  mainly  the  seat  of  the  power  loss  by  hysteresis, 
eddies,  etc.,  which  is  represented  by  the  additional  effective 
resistance,  r",  this  resistance,  r",  is  entirely  or  mainly  assigned  to 
this  circuit. 

In  general,  it  therefore  may  be  assumed: 


x0  =  -, 


Xl  = 


ri  = 


ALTERNATING-CURRENT  TRANSFORMER       207 

Usually,  the  excitation  test  is  made  on  the  low-voltage  coil, 
the  impedance  test  on  the  high-voltage  coil,  and  then  reduced 
to  the  same  coil  as  primary.  Hereby  the  currents  and  voltages 
are  more  nearly  of  the  same  magnitude  in  both  tests. 

154.  In  the  calculation  of  the  transformer: 

The  exciting  admittance,  Fo,  is  derived  by  calculating  the 
total  exciting  current  from  the  ampere-turns  excitation,  the  mag- 
netic characteristic  of  the  iron  and  the  dimensions  of  the  main 
magnetic  circuit,  that  is  the  magnetic  circuit  interlinked  with 
primary  and  secondary  coils.  The  conductance,  gro,  is  derived 
from  the  hysteresis  loss  in  the  iron,  as  given  by  magnetic  density, 
hysteresis  coefficient  and  dimensions  of  magnetic  circuit,  allow- 
ance being  made  for  eddy  currents  in  the  iron. 

The  ohmic  resistances,  r0  and  n,  are  found  from  the  dimen- 
sions of  the  electric  circuit,  and,  where  required,  allowance  made 
for  the  additional  effective  resistance,  r". 

The  reactances,  XQ  and  Xi,  are  calculated  by  calculating  the 
leakage  flux,  that  is  the  magnetic  flux  produced  by  the  total 
primary  respectively  secondary  ampere-turns,  and  passing  be- 
tween primary  and  secondary  coils,  and  within  the  primary 
respectively  secondary  coil,  in  a  magnetic  circuit  consisting 
largely  of  air.  In  this  case,  the  iron  part  of  the  magnetic  leakage 
circuit  can  as  a  rule  be  neglected. 


CHAPTER  XVIII 
POLYPHASE  INDUCTION  MOTORS 

155.  The  induction  motor  consists  of  a  magnetic  circuit  inter- 
linked with  two  electric  circuits  or  sets  of  circuits,  the  primary 
and  the  secondary.  It  therefore  is  electromagnetically  the  same 
structure  as  the  transformer.  The  difference  is,  that  in  the 
transformer  secondary  and  primary  are  stationary,  and  the 
electromagnetic  induction  between  the  circuits  utilized  to  trans- 
mit electric  power  to  the  secondary,  while  in  the  induction  motor 
the  secondary  is  movable  with  regards  to  the  primary,  and  the 
mechanical  forces  between  the  primary,  and  secondary  utilized 
to  produce  motion.  In  the  general  alternating-current  trans- 
former or  frequency  converter  we  shall  find  an  apparatus  trans- 
mitting electric  as  well  as  mechanical  energy,  and  comprising 
both,  induction  motor  and  transformer,  as  the  two  limiting 
cases.  In  the  induction  motor,  only  the  mechanical  fprce  be- 
tween primary  and  secondary  is  used,  but  not  the  transfer  of 
electrical  energy,  and  thus  the  secondary  circuits  are  closed  upon 
themselves.  Hence  the  induction  motor  consists  of  a  magnetic 
circuit  interlinked  with  two  electric  circuits  or  sets  of  circuits, 
the  primary  and  the  secondary  circuit,  which  are  movable  with 
regard  to  each  other.  In  general  a  number  of  primary  and  a 
number  of  secondary  circuits  are  used,  angularly  displaced  around 
the  periphery  of  the  motor,  and  containing  e.m.fs.  displaced  in 
phase  by  the  same  angle.  This  multi-circuit  arrangement  has 
the  object  always  to  retain  secondary  circuits  in  inductive  rela- 
tion to  primary  circuits  and  vice  versa,  in  spite  of  their  relative 
motion. 

The  result  of  the  relative  motion  between  primary  and 
secondary  is,  that  the  e.m.fs.  generated  in  the  secondary  or  the 
motor  armature  are  not  of  the  same  frequency  as  the  e.m.fs. 
impressed  upon  the  primary,  but  of  a  frequency  which  is  the 
difference  between  the  impressed  frequency  and  the  frequency 
of  rotation,  or  equal  to  the  "slip,"  that  is,  the  difference  between 
synchronism  and  speed  (in  cycles). 

208 


POLYPHASE  INDUCTION  MOTORS  209 

Hence,  if 

/  =  frequency  of  main  or  primary  e.m.f., 
s  =  slip  as  fraction  of  synchronous  speed, 
sf  =  frequency  of  armature  or  secondary  e.m.f., 

and  (1  —  s)  /  =  frequency  of  rotation  of  armature. 

In  its  reaction  upon  the  primary  circuit,  however,  the  arma- 
ture current  is  of  the  same  frequency  as  the  primary  current, 
since  it  is  carried  around  mechanically,  with  a  frequency  equal 
to  the  difference  between  its  own  frequency  and  that  of  the 
primary.  Or  rather,  since  the  reaction  of  the  secondary  on  the 
primary  must  be  of  primary  frequency  —  whatever  the  speed 
of  rotation  —  the  secondary  frequency  is  always  such  as  to 
give  at  the  existing  speed  of  rotation  a  reaction  of  primary 
frequency. 

156.  Let  the  primary  system  consist  of   p0   equal   circuits, 

displaced  angularly  in  space  by  —  of  a  period,  that  is,  —  of 

PQ  PO 

the  width  of  two  poles,  and  excited  by  p0  e.m.fs.  displaced  in 
phase  by  —  of  a  period;  that  is,  in  other  words,  let  the  field 

circuits    consist    of    a    symmetrical    po-phase   system.     Analo- 
gously, let  the  armature  or  secondary  circuits  consist  of  a  sym- 
metrical pi-phase  system. 
Let 

n0  -=  number  of  primary  turns  per  circuit  or  phase; 
n\  =  number  of  secondary  turns  per  circuit  or  phase; 


Pi 

Since  the  number  of  secondary  circuits  and  number  of  turns 
of  the  secondary  circuits,  in  the  induction  motor  —  as  in  the 
stationary  transformer  —  is  entirely  unessential,  it  is  preferable 
to  reduce  all  secondary  quantities  to  the  primary  system,  by  the 
ratio  of  transformation,  a;  thus 

if  E'i  =  secondary  e.m.f.  per  circuit, 

EI  =  aE'i  =  secondary  e.m.f.  per  circuit  reduced  to  primary 
system  ; 

14 


210         ALTERNATING-CURRENT  PHENOMENA 

if  I'  i  =  secondary  current  per  circuit, 

j/ 
/i  =  -^  =  secondary  current  per  circuit  reduced  to  primary 

system; 
if  r'i  =  secondary  resistance  per  circuit, 

TI  =  azbr'i  =  secondary  resistance  per  circuit  reduced  to  pri- 

mary system; 
if  x'i  =  secondary  reactance  per  circuit, 

Xi  =  a2bx'i  =  secondary  reactance  per  circuit  reduced  to  pri- 

mary system; 
if  z'i  =  secondary  impedance  per  circuit, 

Zi  —  a?bz'i  =  secondary  impedance  per  circuit  reduced  to  pri- 
mary system; 

that  is,  the  number  qf  secondary  circuits  and  of  turns  per  sec- 
ondary circuit  is  assumed  the  same  as  in  the  primary  system. 

In  the  following  discussion,  as  secondary  quantities,  the 
values  reduced  to  the  primary  system  shall  be  exclusively 
used,  so  that,  to  derive  the  true  secondary  values,  these  quan- 
tities have  to  be  reduced  backward  again  by  the  factor 


157.  Let 

«f>  =  total  maximum  flux  of  the  magnetic  field  per  motor  pole.    • 
We  then  have  '  . 

E  =  \/2  wnof  3>  10~8  =  effective  e.m.f  .  generated  by  the  magnetic 
field  per  primary  circuit. 

Counting  the  time  from  the  moment  where  the  rising  mag- 
netic flux  of  mutual  induction,  $  (flux  interlinked  with  both 
electric  circuits,  primary  and  secondary),  passes  through  zero, 
in  complex  quantities,  the  magnetic  flux  is  denoted  by 

$  =  —  j&, 
and  the  primary  generated  e.m.f., 

E-  -e; 
where 
e  =  \/2  TrnfQ  10~8  may  be  considered  as  the  "active  e.m.f.  of  the 

motor,"  or  "counter  e.m.f." 

Since  the  secondary  frequency  is  sf,  the  secondary  induced 
e.m.f.  (reduced  to  primary  system)  is  EI  =  —  se. 


POLYPHASE  INDUCTION  MOTORS  211 

Let 
70  =  exciting  current,  or  current  through  the  motor,  per  primary 

circuit,  when  doing  no  work  (at  synchronism), 
and 

F  .=  g  —  jb  =  primary  exciting  admittance  per  circuit  =  — * 

• .  6 

We  thus  have, 

ge  =  magnetic  power  current,  ge2  =  loss  of  power  by  hysteresis 
(and  eddy  currents)  per  primary  coil. 

Hence 
p0ge2  =  total  loss  of  power  by  hysteresis  and  eddies,  as  calculated 

accorbing  to  Chapter  XII. 
be  —  magnetizing  current,  and 
nQbe  =  effective  m.m.f.  per  primary  circuit; 

hence 

~nobe  =  total  effective  m.m.f., 

a 

and 

nr\  - 

T^nobe  =  total  maximum  m.m.f.,  as  resultant  of  the  m.m.f s. 

of  the  po-phases,  combined  by  the  parallelogram  of 
m.m.fs.1 

If  (R  =  reluctance  of  magnetic  circuit  per  pole,  as  discussed 
in  Chapter  XII,  it  is 


V2 

Thus,  from  the  hysteretic  loss,  and  the  reluctance,  the  con- 
stants, g  and  b  and  thus  the  admittance,  F,  are  derived. 

Let  r0  =  resistance  per  primary  circuit; 
XQ  =  reactance  per  primary  circuit; 

thus, 

Z0  =  r0  +  JXQ  =  impedance  per  primary  circuit ; 

7*1  =  resistance  per  secondary  circuit  reduced  to  primary 

system; 
Xi  =  reactance  per  secondary  circuit  reduced  to  primary 

system,  at  full  frequency  /; 

1  Complete  discussion  hereof,  see  Chapter  XXXIII. 


212         ALTERNATING-CURRENT  PHENOMENA 

hence, 

sxi  =  reactance  per  secondary  circuit  at  slip  s, 
and 

Zi  =  r\  +  jsxi  =  secondary  internal  impedance. 

168.  We  now  have, 
Primary  generated  e.m.f., 

E  =  -e. 
Secondary  generated  e.m.f., 

Ei  =  —  se. 
Hence, 
Secondary  current, 


Component   of   primary   current,    corresponding   thereto,    or 
primary  load  current, 


Primary  exciting  current, 

/o=  eY  =  e  (g  —  jb);  hence, 
Total  primary  current, 
7  =  /'  +  Jo 


e.m.f.  consumed  by  primary  impedance, 
E,  =  Z07 


e.m.f.  required  to  overcome  the  primary  generated  e.m.f., 

-E  =  e; 
hence, 

Primary  terminal  voltage, 
Eo  =  e  +  E. 


We  get  thus,  in  an  induction  motor,  at  slip  s  and  active  e.m.f.  e, 
Primary  terminal  voltage, 


POLYPHASE  INDUCTION  MOTORS  213 

Primary  current, 


or,  in  complex  expression, 
Primary  terminal  voltage, 


E0  =  e 
Primary  current, 

7  -|£  +  r). 

I  £  1 

To  eliminate  e,  we  divide,  and  get, 

Primary  current,  at  slip  s,  and  impressed  e.m.f.,  #0; 


r  s         i 

" 


or 


(flf  - 


s  (r0  +  J30)  +  (r0 
Neglecting,  in  the  denominator,  the  small   quantity   Z0ZiF, 
it  is 


sxi)  (g  - 


=  (s  +  nflf  +  sxib)  —  j  (rib  -  s 
(ri  +  sr0)  +  js  (xi  +  XQ) 
or,  expanded, 

s2r0)  +  n2g  +  sri  (r0g  -  xQb)  +  s2X 


j[s2  (0 


Hence,   displacement  of  phase  between  current  and  e.m.f., 


s2r0)  -f  r!2fif  +  sr,(r0flf  -  ^ 
Neglecting  the  exciting  current,  7o,  altogether,  that  is,  setting 
Y  =  0, 
We  have 

7  _     „ 
= 


(ri  +  sr0)  +  js  (x0  4-  x^ ' 

S  (X0  +  Xi) 

tan   00  = r 


214         ALTERNATING-CURRENT  PHENOMENA 

159.  In  graphic  representation,  the  induction  motor  diagram 
appears  as  follows: — 

Denoting  the  magnetism  by  the  vertical  vector  0$  in  Fig.  118, 
the  m.m.f.  in  ampere-turns  perjcircuit  is  represented  by  vector 
OF,  leading  the  magnetism,  0$,  by  the  angle  of  hysteretic 
advance  a.  The  e.m.f.  generated  in  the  secondary  is  propor- 
tional to  the  slip  s,  and  represented  by  OEi  at  the  amplitude  of 
180°.  Dividing  OE\  by  a  in  the  proportion  of  r\  -*•  sx\,  and 
connecting  a  with  the  middle,  b,  of  the  lower  arc  of  the  circle, 
OEi,  this  line  intersects  the  upper  arc  of  the  circle  at  the  point, 
IiTi.  Thus,  0/iri  is  the  e.m.f.  consumed  by  the  secondary 
resistance,  and  OI\x\  equal  and  parallel  to  E\I\ri  is  the  e.m.f. 
consumed  by  the  secondary  reactance.  The  angle,  EiOItfi  =  61, 
is  the  angle  of  secondary  lag. 


FIG.  118. 

The  secondary  m.m.f.,  OGi,  is  in  the  direction  of  the  vector, 
OIiTi.  Completing  the  parallelogram  of  m.m.fs.  with  OF  as 
diagonal  and  OGi,  as  one  side,  gives  the  primary  m.m.f.,  OG, 
as  other  side.  The  primary  current  and  the  e.m.f.  consumed 
by_the  primary  resistance,  represented  by  OIrQ,  is  in  line  with 
OG,_the  e.m.f.  consumed  by  the  primary  reactance  90°  ahead 
of  OG,  and  represented  by  OIxQ,  and  their  resultant,  OIzQ,  is  the 
e.m.f.  consumed  by  the  primary  impedance.  The  e.m.f.  gener- 
ated in  the  primary  circuit  is  OEf,  and  the  e.m.f.  required  to 
overcome  this  counter  e.m.f.  is  QE  equal  and  opposite  to  OE'. 
Combining  OE  with  OIzQ  gives  the  primary  terminal  voltage 
represented  by  vector  OE0)  and  the  angle  of  primary  lag, 
EoOG  —  &Q. 


POLYPHASE  INDUCTION  MOTORS 


215 


160.  Thus  far  the  diagram  is  essentially  the  same  as  the 
diagram  of  the  stationary  alternating-current  transformer.  Re- 
garding dependence  upon  the  slip  of  the  motor,  the  locus  of 
the  different  quantities  for  different  values  of  the  slip,  s,  is 
determined  thus, 


FIG.  119. 


Let 


Assume  in  opposition  to  0$,  a  point,  A,  such  that 
.  OA  -5-  I\r\  =  EI  -T-  IiSXi,  then 
X  #1       Iin  X  s#' 


OA  = 


IlSXi 


E' 


constant. 


That  is,  Itfi  lies  on  a  half-circle  with  OA  —  —  E'  as  diameter. 

That  means  GI  lies  on  a  half -circle,  gif  in  Fig.  119  with  OC 
as  diameter.  In  consequence  hereof,  GQ  lies  on  half-circle  g0 
with  FB  equal  and  parallel  to  OC  as  diameter. 


216        ALTERNATING-CURRENT  PHENOMENA 

Thus  7r0  lies  on  a  half-circle  with  DH  as  diameter,  which 
circle  is  perspective  to  the  circle,  FB,  and  IxQ  lies  on  a  half- 
circle  with  IK  as  diameter,  and  Iz0  on  a  half-circle  with  LN 
as  diameter,  which  circle  is  derived  by  the  combination  of  the 
circles,  7r0  and  Ix0. 

The  primary  terminal  voltage,  E0,  lies  thus  on  a  half-circle, 
e0,  equal  to  the  half-circle,  7z0,  and  having  to  point  E  the  same 
relative  position  as  the  half-circle,  7z0,  has  to  point  0. 

This  diagram  corresponds  to  constant  intensity  of  the  maxi- 
mum magnetism,  0$.  If  the  primary  impressed  voltage,  E0, 
is  kept  constant,  the  circle,  e0,  of  the  primary  impressed  voltage 
changes  to  an  arc  with  0  as  center,  and  all  the  corresponding 
points  of  the  other  circles  have  to  be  reduced  in  accordance 
herewith,  thus  giving  as  locus  of  the  other  quantities  curves  of 
higher  order  which  most  conveniently  are  constructed  point  for 
point  by  reduction  from  the  circle  of  the  loci  in  Fig.  119. 

Torque  and  Power 

161.  The  torque   developed   per  pole  by  an  electric   motor 

$ 
equals  the  product  of  effective  magnetism,  —  /=,  times  effective 

v  2 

F 

armature  m.m.f.,  —  7=,  times  the  sine  of  the  angle  between  both, 
V2 

&F 
IX  =~  sin  (*F). 

If  tt]  =  number  of  turns,  7i  =  current,  per  circuit,  with  pi 
armature  circuits,  the  total  maximum  current  polarization,  or 
m.m.f.  of  the  armature,  is 


Hence  the  torque  per  pole, 


If  q  =  the  number  of  poles  of  the  motor,  the  total  torque  of 
the  motor  is, 

P.; 


POLYPHASE  INDUCTION  MOTORS  217 

The  secondary  induced  e.m.f.,  EI,  lags  90°  behind  the  inducing 
magnetism,  hence  reaches  a  maximum  displaced  in  space  by 
90°  from  the  position  of  maximum  magnetization.  Thus,  if 
the  secondary  current,  7i,  lags  behind  its  emf.,  EI,  by  angle, 
0i,  the  space  displacement  between  armature  current  and  field 
magnetism  is 

£  (/i*)  =  90°  +  0!, 

hence  sin  ($/i)  =  cos  0i. 

We  have,  however, 

cos  0i  =      ,— ri     =  , 

vV  +  W 

es  10-1 


+ 

6  =  V2" 
thus, 


substituting  these  values  in  the  equation  of  the  torque,  it  is 

qp^sr^  107 

jj     —    • 

or,  in  practical  (c.g.s.)  units, 


is  the  torque  of  the  induction  motor. 

At  the  slip,  s,  the  frequency,  /,  and  the  number  of  poles,  q} 
the  linear  speed  at  unit  radius  is 


hence  the  output  of  the  motor, 

P  =  Dv, 
or,  substituted, 

Pirie28  (1  -  a) 

"     .     n2  +  S2*!2 

is  </ie  power  of  the  induction  motor. 

162.  We  can  arrive  at  the  same  results  in  a  different  way: 
By  the  counter  e.m.f.,  e,  of  the  primary  circuit  with  current 
/  =  IQ  +  1  1  the  power  is  consumed,  el  =  e!Q  +  eli.    The  power, 
6/0,  is  that  consumed  by  the  primary  hysteresis  and  eddys. 


218         ALTERNATING-CURRENT  PHENOMENA 

The  power,  el\,  disappears  in  the  primary  circuit  by  being 
transmitted  to  the  secondary  system. 

Thus  the  total  power  impressed  upon  the  secondary  system, 
per  circuit,  is 

Pi  =  el,. 

Of  this  power  a  part,  EJi,  is  consumed  in  the  secondary  circuit 
by  resistance.  The  remainder, 

disappears  as  electrical  power  altogether;  hence,  by  the  law 
of  conservation  of  energy,  must  reappear  as  some  other  form  of 
energy,  in  this  case  as  mechanical  power,  or  as  the  output  of 
the  motor  (including  friction). 

Thus  the  mechanical  output  per  motor  circuit  is 

P'  =  /!  (e  -  E,). 
Substituting, 

EI  =  se; 

'T  -        se     - 

J.  i   ^^    .  -  -  -  -  • 

it  is 

pt    =   6_2S  (1  -  8) 


2     |        2       2 

hence,  since  the  imaginary  part  has  no  meaning  as  power, 
P*  = —  * 

and  the  total  power  of  the  motor, 

p  =  pirie2s  (1  -  s)_ 

At  the  linear  speed, 
at  unit  radius  the  torque  is 


In  the  foregoing,  we  found 

Eo  =  e  1 1  +  s  ^  +  Z0Y  } 

(  A\  J 


POLYPHASE  INDUCTION  MOTORS  219 

or,  approximately, 

Eo  =  e  { 1  +  s  |-° }  ; 
or, 

g    =      ! 

expanded, 

6   ==   Mi  Q  ~/         j — 

••     v  i    \~  $7*0) 

or,  eliminating  imaginary  quantities, 


e  = 


Substituting  this   value  in   the  equations   of  torque  and   of 
power,  they  become, 
torque, 


power, 


Maximum  Torque 

163.  The  torque  of  the  induction  motor  is  a   maximum  for 
that  value  of  slip,  s,  where 


or,  since 

D  = 

for, 

"   !  v  i    i    •"•  u/     i    •"   v*" i    i   «^u/    '     _  n 

dst  ~7~  ~J  :=U; 

expanded,  this  gives, 

n2        2  2  _ 

s2  ' 

or, 

st  = 


Vro2  +  (^i  +  x0)2 


220         ALTERNATING-CURRENT  PHENOMENA 

Substituting  this  in  the  equation  of  torque,  we  get  the  value  of 
maximum  torque, 

A  = - 

that  is,  independent  of  the  secondary  resistance,  TI. 

The  power  corresponding  hereto  is,  by  substitution  of  st  in  P, 


Pt  = 


2\/r02  4- 


This  power  is  not  the  maximum  output  of  the  motor,  but 
is  less  than  the  maximum  output.  The  maximum  output  is 
found  at  a  lesser  slip,  or  higher  speed,  while  at  the  maximum 
torque  point  the  output  is  already  on  the  decrease,  due  to  the 
decrease  of  speed. 

With  increasing  slip,  or  decreasing  speed,  the  torque  of  the 
induction  motor  increases;  or  inversely,  with  increasing  load, 
the  speed  of  the  motor  decreases,  and  thereby  the  torque  in- 
creases, so  as  to  carry  the  load  down  to  the  slip,  st,  correspond- 
ing to  the  maximum  torque.  At  this  point  of  load  and  slip 
the  torque  begins  to  decrease  again;  that  is,  as  soon  as  with 
increasing  load,  and  thus  increasing  slip,  the  motor  passes  the 
maximum  torque  point,  sh  it  "falls  out  of  step,"  and  comes  to  a 
standstill. 

Inversely,  the  torque  of  the  motor,  when  starting  from  rest, 
increases  with  increasing  speed,  until  the  maximum  torque 
point  is  reached.  From  there  toward  synchronism  the  torque 
decreases  again. 

In  consequence  hereof,  the  part  of  the  torque-speed  curve 
below  the  maximum  torque  point  is  in  general  unstable,  and  can 
be  observed  only  by  loading  the  motor  with  an  apparatus  whose 
counter-torque  increases  with  the  speed  faster  than  the  torque 
of  the  induction  motor. 

In  general,  the  maximum  torque  point,  st,  is  between  syn- 
chronism and  standstill,  rather  nearer  to  synchronism.  Only 
in  motors  of  very  large  armature  resistance,  that  is,  low  efficiency, 
st  >  1,  that  is,  the  maximum  torque,  occurs  below  standstill, 
and  the  torque  constantly  increases  from  synchronism  down 
to  standstill. 

It  is  evident  that  the  position  of  the  maximum  torque  point, 
st,  can  be  varied  by  varying  the  resistance  of  the  secondary 


POLYPHASE  INDUCTION  MOTORS  221 

circuit,  or  the  motor  armature.  Since  the  slip  of  the  maxi- 
mum torque  point,  st,  is  directly  proportional  to  the  armature 
resistance,  ri,  it  follows  that  very  constant  speed  and  high 
efficiency  brings  the  maximum  torque  point  near  synchronism, 
and  gives  small  starting  torque,  while  good  starting  torque 
means  a  maximum  torque  point  at  low  speed;  that  is,  a  motor 
with  poor  speed  regulation  and  low  efficiency. 

Thus,  to  combine  high  efficiency  and  close  speed  regulation 
with  large  starting  torque,  the  armature  resistance  has  to  be 
varied  during  the  operation  of  the  motor,  and  the  motor  started 
with  high  armature  resistance,  and  with  increasing  speed  this 
armature  resistance  cut  out  as  far  as  possible. 

164.  If 

«*  =  1, 

ri  =  Vr02+  Oi+Zo)2. 

In  this  case  the  motor  starts  with  maximum  torque,  and  when 
overloaded  does  not  drop  out  of  step,  but  gradually  slows  down 
more  and  more,  until  it  comes  to  rest. 
If 

St    >   1, 

then 


In  this  case,  the  maximum  torque  point  is  reached  only  by 
driving  the  motor  backward,  as  counter-torque. 

As  seen  above,  the  maximum  torque,  Dh  is  entirely  inde- 
pendent of  the  armature  resistance,  and  likewise  is  the  current 
corresponding  thereto,  independent  of  the  armature  resistance. 
Only  the  speed  of  the  motor  depends  upon  the  armature  resistance. 

Hence  the  insertion  of  resistance  into  the  motor  armature 
does  not  change  the  maximum  torque,  and  the  current  corre- 
sponding thereto,  but  merely  lowers  the  speed  at  which  the 
maximum  torque  is  reached. 

The  effect  of  resistance  inserted  into  the  induction  motor  is 
merely  to  consume  the  e.m.f.,  which  otherwise  would  find  its 
mechanical  equivalent  in  an  increased  speed,  analogous  to 
resistance  in  the  armature  circuit  of  a  continuous-current  shunt 
motor. 

Further  discussion  on  the  effect  of  armature  resistance  is 
found  under  "Starting  Torque." 


222         ALTERNATING-CURRENT  PHENOMENA 

Maximum  Power 

165.  The  power  of  an  induction  motor  is  a  maximum  for  that 
slip,  sp,  where 

dp      fi- 
ds       °> 

or,  since 

(l  -  *) 


ds 
expanded,  this  gives 


r>   _   

"  (n  -f  sr0r  -1-  s\Xi  -f  z0r 

I      <fj.  ^  2  _j_   S2  ('/j.        I     /^.  "^2  i 

=  0; 


fi 

substituted  in  P,  we  get  the  maximum  power, 

P    — 

•t      71 


r0)  +  \Of,  +  r0)2  +  (*i  +  *0)2} 

This  result  has  a  simple  physical  meaning:  (ri  +  r0)  =  r  is 
the  total  resistance  of  the  motor,  primary  plus  secondary  (the 
latter  reduced  to  the  primary),  (xi  +  XQ)  is  the  total  reactance, 


and   thus  V(n  +  r0)2  +  (xi  +  x0)2  =  z  is  the  total  impedance 
of  the  motor.     Hence 


2 

p  /'i-'O 

^P    ~i 


is  the  maximum  output  of  the  induction  motor,  at  the  slip, 


The  same  value  has  been  derived  in  Chapter  X,  as  the  maxi- 
mum power  which  can  be  transmitted  into  a  non-inductive 
receiver  circuit  over  a  line  of  resistance,  r,  and  impedance,  z, 
or  as  the  maximum  output  of  a  generator,  or  of  a  stationary 
transformer.  Hence: 

The  maximum  output  of  an  induction  motor  is  expressed  by 
the  same  formula  as  the  maximum  output  of  a  generator,  or  of  a 
stationary  transformer,  or  the  maximum  output  which  can  be 
transmitted  over  an  inductive  line  into  a  non-inductive  receiver 
circuit. 


POLYPHASE  INDUCTION  MOTORS  223 

The  torque  corresponding  to  the  maximum   output,  Pp,  is 


This  is  not  the  maximum  torque;  but  the  maximum  torque, 
Dt,  takes  place  at  a  lower  speed,  that  is,  greater  slip, 


Vr02  + 
since 


ri-fVOi+r0)2  + 
that  is, 

st  >  sp. 

It  is  obvious  from  these  equations,  that,  to  reach  as  large 
an  output  as  possible,  r  and  z  should  be  as  small  as  possible; 
that  is,  the  resistances,  ri  +  r0,  and  the  impedances,  z,  and  thus 
the  reactances,  x\  +  XQ,  should  be  small.  Since  ri  +  r0  is 
usually  small  compared  with  xi  -f-  XQ  it  follows,  that  the  problem 
of  induction  motor  design  consists  in  constructing  the  motor  so 
as  to  give  the  minimum  possible  reactances,  x\  +  XQ. 

Starting  Torque 
166.  In  the  moment  of  starting  an  induction  motor,  the  slip  is 

s  =  1; 
hence,  starting  current, 

0*1  +jxi)  +  0*0  -h  jxo)  +  0*i  +jxi)  +  0"o  +jxo)  (g  —  jb)     °' 


or,  expanded,  with  the  rejection  of  the  last  term  in  the  denomi- 
nator, as  insignificant, 


r0)  +  0(ri[ri  +  r0]  +  x^Xi  +  XQ]) 

r0]  -f  xi[xi  +  XQ])  -  g 
r0) 


T  _        i        0  ii        0         ii        Q  -, 

2 


and,  displacement  of  phase,  or  angle  of  lag, 

_  (xi  +  XQ)  +  b  (rt  [ri  +  r0] 

~ 


r0)  +  g  0*1  [ri  +  r0] 


224         ALTERNATING-CURRENT  PHENOMENA 

Neglecting  the  exciting  current,  g  =  0  =  b,  these  equations 
assume  the  form, 


7 
= 


r0)2  +  (xi  +  *o)2  •         (rt  +  r0) 
or,  eliminating  imaginary  quantities, 


=  •  =        Q 

~  V(ri  +  r0)2  +  (zi  +  *o)2    :   *  ; 
and 

.     Xi+X0 

tan  0n  -  ;  -- 
T\  +  r0 

That  means,  that  in  starting  the  induction  motor  without 
additional  resistance  in  the  armature  circuit  —  in  which  case 
Xi  +  XQ  is  large  compared  with  r\  +  r0,  and  the  total  impe- 
dance, z,  small  —  the  motor  takes  excessive  and  greatly  lagging 
currents. 

The  starting  torque  is 


47T/       Z2* 

That  is,  the  starting  torque  is  proportional  to  the  armature 
resistance,  and  inversely  proportional  to  the  square  of  the  total 
impedance  of  the  motor. 

It  is  obvious  thus,  that,  to  secure  large  starting  torque,  the 
impedance  should  be  as  small,  and  the  armature  resistance  as 
large,  as  possible.  The  former  condition  is  the  condition  of 
large  maximum  output  and  good  efficiency  and  speed  regula- 
tion; the  latter  condition,  however,  means  inefficiency  and  poor 
regulation,  and  thus  cannot  properly  be  fulfilled  by  the  internal 
resistance  of  the  motor,  but  only  by  an  additional  resistance 
which  is  short-circuited  while  the  motor  is  in  operation. 

Since,  necessarily, 

n  <  z, 
we  have, 


and  since  the  starting  current  is,  approximately 


POLYPHASE  INDUCTION  MOTORS  225 

we  have, 


-L/00    =    ~A r-C'O-t 

47T/ 

would   be  the  theoretical   torque   developed   at   100  per  cent, 
efficiency  and  power-factor,   by  e.m.f.  E0,  and  current  /,   at 
synchronous  speed. 
Thus, 

D0  <  Doo, 

and  the  ratio  between  the  starting  torque,  Do,  and  the  theo- 
retical maximum  torque,  Doo,  gives  a  means  to  judge  the  per- 
fection of  a  motor  regarding  its  starting  torque. 

DQ 

This  ratio,  yy-,  exceeds  0.9  in  the  best  motors. 

•pi 
Substituting  I  =  —  in  the  equation  of  starting  torque,  it 

assumes  the  form, 


Since  — -  =  synchronous  speed,  it  is: 

'  The  starting  torque  of  the  induction  motor  is  equal  to  the  resistance 
loss  in  the  motor  armature,  divided  by  the  synchronous  speed. 

The  armature  resistance  which  gives  maximum  starting  torque 
is 

"  =  0 


dn 
or  since, 

Do  = 


4vr/    (r!  +  r0)2  +  ( 

T~  ~^~~  \   =  0> 

dri  (  ri 


expanded,  this  gives, 


the  same  value  as  derived  in  the  paragraph  on   "maximum 
torque." 

Thus,  adding  to  the  internal  armature  resistance,  r'i,  in  start- 
ing the  additional  resistance, 


15 


226 


ALTERNATING-CURRENT  PHENOMENA 


makes  the  motor  start  with  maximum  torque,  while  with 
increasing  speed  the  torque  constantly  decreases,  and  reaches 
zero  at  synchronism.  Under  these  conditions,  the  induction 
motor  behaves  similarly  to  the  continuous-current  series  motor, 
varying  in  speed  with  the  load,  the  difference  being,  however, 
that  the  induction  motor  approaches  a  definite  speed  at  no-load, 
while  with  the  series  motor  the  speed  indefinitely  increases  with 
decreasing  load. 


10 


20 


so 


40 


GO 


70 


90 


100?. 


20  H.P.  THREE-PHASE  INDUCTION  MOTOR 
110  VOLTS,  900  REVOLUTIONS,  6O  CYCLES 
Y  =  .1-.4J     rl=.02/.045/.18  /  .75 
Z0=.03  +  .09j 
Z,=.02  +  -085J 


0  10         20          30          40  50         60          70          80         90         100 

SPEED,  PER  CENT  SYNCHRONISM 

0     20  40   60   80  100  120  140 160  180  200  220  240  260  280  300  320  340 
AMPERES 

FIG.  120. 

The  additional  armature  resistance,  r"i,  required  to  give 
a  certain  starting  torque,  is  found  from  the  equation  of  starting 
torque : 

Denoting  the  internal  armature  resistance  by  r'i,  the  total 
armature  resistance  is  TI  =  r'i  -f-  r"i, 
and  thus, 


qpiE( 


r"i 


hence, 


POLYPHASE  INDUCTION  MOTORS  227 

This  gives  two  values,  one  above,  the  other  below,  the  maxi- 
mum torque  point. 

Choosing  the  positive  sign  of  the  root,  we  get  a  larger  armature 
resistance,  a  small  current  in  starting,  but  the  torque  constantly 
decreases  with  the  speed. 

Choosing  the  negative  sign,  we  get  a  smaller  resistance,  a 
large  starting  current,  and  with  increasing  speed  the  torque 
first  increases,  reaches  a  maximum,  and  then  decreases  again 
toward  synchronism. 

These  two  points  correspond  to  the  two  points  of  the  speed- 
torque  curve  of  the  induction  motor,  in  Fig.  120,  giving  the 
desired  torque,  D0. 

The  smaller  value  of  r"\  gives  fairly  good  speed  regulation, 
and  thus  in  small  motors,  where  the  comparatively  large  start- 
ing current  is  no  objection,  the  permanent  armature  resistance 
may  be  chosen  to  represent  this  value. 

The  larger  value  of  r"\  allows  to  start  with  minimum  current', 
but  requires  cutting  out  of  the  resistance  after  the  start,  to 
secure  speed  regulation  and  efficiency. 

167.  Approximately,  the  torque  of  the  induction  motor  at 
any  slip,  s: 

D  =  —-^-, r 


can  be  expressed  in  a  simple  and  so  convenient  form  as  function 
of  the  maximum  torque: 

Dt  =  c 


or  of  the  starting  torque:  s  =  1  : 


Dividing  D  by  Dt  we  have 


D  =  0i         on 

(ri  +  sr0)2  +  s2  (xl  +  *0)2 

Since  r0,.  the  primary  resistance,  is  small  compared  with 
x  =  xi  +  XQ, 


228         ALTERNATING-CURRENT  PHENOMENA 

the  total  self-inductive  reactance  of  the  motor,  it  can  be  neg 
lected  under  the  square  root,  and  the  equation  so  gives  : 

= 


or,  still  more  approximately: 

- 


and  the  starting  torque,  f  or  :  s  =  1  : 


hence,  dividing, 


D  ._  «  (ri2  +  x*) 

"    " 


or,  if  7*1  is  small  compared  with  x,  that  is,  in  a  motor  of  low 
resistance  armature: 


sx2 

D   =        o ;       — o  DQ, 


From  the  equation: 


it  follows  that  for  small  values  of  s,  or  near  synchronism: 


by  neglecting  s2x2  compared  with  ri2: 

For  low  values  of  speed,  or  high  values,  of  s,  it  follows,  by 
neglecting  ri2  compared  with  s2x2: 


that  is,  approximately,  near  synchronism,  the  torque  is  directly 
proportional  to  the  slip,  and  inversely  proportional  to  the 
armature  resistance,  that  is,  proportional  to  the  ratio 

—  r—  -  ;  near  standstill,  the  torque  is  inversely  pro- 
armature  resistance' 

portional  to  the  slip,  but  directly  proportional  to  the  armature 
resistance,  and  so  is  increased  by  increasing  the  armature  resist- 
ance in  a  motor  of  low-armature  resistance. 


POLYPHASE  INDUCTION  MOTORS  229 

Synchronism 

168.  At  synchronism,  s  =  0,  we  have, 
Ia  =  E0(g  -jb); 


or, 


p  =  o,  D  =  0; 

that  is,  power  and  torque  are  zero.  Hence,  the  induction 
motor  can  never  reach  complete  synchronism,  but  must  slip 
sufficiently  to  give  the  torque  consumed  by  friction. 

Running  near  Synchronism 

169.  When  running  near  synchronism,  at  a  slip,  s,  above 
the  maximum  output  point,  where  s  is  small,  from  0.01  to  0.05 
at  full-load,  the  equations  can  be  simplified  by  neglecting  terms 
with  s,  as  of  higher  order. 

We  then  have,  current, 

T       s  +  n  (fir  -  jb) 

1  -     yr   ~  Eo' 

or,  eliminating  imaginary  quantities, 


angle  of  lag, 

tan  0o  = 


_ 

: 
or,  inversely, 


s  = 
that  is, 


S2  (X\  -f  X0)  +  Ti2b 

r1 

"^"PJV. 

s  +  rig 

synchronism,  the  slip,  s,  of  an  induction  motor,  or  its  drop 


230 


ALTERNATING-CURRENT  PHENOMENA 


in  speed,  is  proportional  to  the  armature  resistance,  r\,  and  to  the 
power,  P,  or  torque,  D. 


EXAMPLE 


170.  As  an  example  are  shown,  in  Fig.  120,  characteristic 
curves  of  a  20-hp.  three-phase  induction  motor,  of  900  revolutions 
synchronous  speed,  8  poles,  frequency  of  60  cycles. 


32 
30 

28 
26 
5  24 

O  22 

a. 

1^20 

CE 
0 

3" 
I12 

a;  10 

I- 

6 
4 
2 

0 

1 

u 

z 

tf> 

u. 

o 

t- 
z 

UJ 

o 

o: 

1. 
Si 

UJ|- 
CLUJ 

O 

s 

100 
90 
80 
70 
60 
50 
40 
30 
20 
10 

•  2( 
110 

5  H.P.  THREE  PHASE 
VOLTS.    900  REVOLl 

CURRENT  DIAGRAM 
Y=.1-.4j 
Z0-.03-»-.09J 

NDUCTION  MOTOR 
JTIONS.  60  CYCLES 

I 

s 

—  *^ 

t\ 

/ 

T( 

)RQL 

JE 

\ 

<^ 

/ 

X 

\' 

\ 

« 

y 

/ 

i 

$- 

7 

/ 

\ 

\\ 

-=: 



7^ 

—  7 

— 

— 

J3 

'ED 

s 

A 

/ 

/ 

"*^ 

~~^-*. 

^c 

fygj 

--, 

\ 

\ 

/ 

/ 

"*^*. 

I 

s 

\ 

\\ 

y 

f               : 

**>. 

\ 

n\ 

/ 

/ 

^•*> 

^>.^ 

la 

I/ 

^"^H  \ 

I 

/ 

I 

1 

I 

1 

I 

/ 

50 


100 


150  200 

AMPERES 

FIG.  121. 


250 


300 


350 


The  impressed  e.m.f.  is  110  volts  between  lines,  and  the  motor 
star  connected,  hence  the  e.m.f.  impressed  per  circuit: 


110 
V3 


=  63.5;  or#0  =  63.5. 


The  constants  of  the  motor  are: 


POLYPHASE  INDUCTION  MOTORS  231 

Primary  admittance,     Y    =  0.1  —  0.4  j. 
Primary  impedance,      Z    =  0.03  +  0.09  j. 
Secondary  impedance,  Z\  —  0.02  +  0.085  j. 

In  Fig.  120  is  shown,  with  the  speed  in  per  cent,  of  synchronism, 
as  abscissas,  the  torque  in  kilogram-meters  as  ordinates  in  drawn 
lines,  for  the  values  of  armature  resistance: 
TI  =  0.02  :  short-circuit  of  armature,  full  speed. 
TI  =  0.045:  0.025  ohms  additional  resistance. 
TI  =  0.18  :  0.16  ohms  additional,  maximum  starting  torque. 
7*1  =  0.75  : 0.73    ohms    additional,    same    starting    torque    as 

n  =  0.045. 


20  H. P.  THREE-PHASE      | 

INDUCTION  MOTOR 
110  VOLTS,  900  REVOLUTIONS 

60  CYCLES 
SPEED  DIAGRAM 

Y  =.1  -  4  j 

Z0=.03  +.09J 
Z1=.045  +  .085j 


Speed,  Percent  of  Synchronism 


FIG.  122. 

On  the  same  figure  is  shown  the  current  per  line,  in  dotted 
lines,  with  the  verticals  or  torque  as  abscissas,  and  the  hori- 
zontals or  amperes  as  ordinates.  To  the  same  current  always 
corresponds  the  same  torque,  no  matter  what  the  speed  may  be. 

On  Fig.  121  is  shown,  with  the  current  input  per  line  as 
abscissas,  the  torque  in  kilogram-meters  and  the  output  in  horse- 
power as  ordinates  in  drawn  lines,  and  the  speed  and  the  mag- 
netism, in  per  cent,  of  their  synchronous  values,  as  ordinates  in 
dotted  lines,  for  the  armature  resistance,  r\  =  0.02,  or  short- 
circuit. 


232         ALTERNATING-CURRENT  PHENOMENA 

In  Fig.  122  is  shown,  with  the  speed,  in  per  cent,  of  synchro- 
nism, as  abscissas,  the  torque  in  drawn  line,  and  the  output  in 
dotted  line,  for  the  value  of  armature  resistance  TI  =  0.045, 
for  the  whole  range  of  speed  from  120  per  cent,  backward 
speed  to  200  per  cent,  beyond  synchronism,  showing  the  two 
maxima,  the  motor  maximum  at  s  =  0.25,  and  the  generator 
maximum  at  s  =  —  0.25. 

171.  As  seen  in  the  preceding,  the  induction  motor  is  charac- 
terized by  the  three  complex  imaginary  constants, 
Yo  =  gQ  —  jbo,  the  primary  exciting  admittance, 

ZQ  =  rQ  +  jxo,  the  primary  self-inductive  impedance,  and 

Zi  =  7*1  +  jxi,  the  secondary  self-inductive  impedance, 
reduced  to  the  primary  by  the  ratio  of  secondary  to  primary 
turns. 

From  these  constants  and  the  impressed  e.m.f.,  eQ,  the  motor 
can  be  calculated  as  follows: 

Let, 

e  =  counter  e.m.f.  of  motor,  that  is,  e.m.f.  generated  in  the 
primary  by  the  mutual  magnetic  flux. 

At  the  slip,  s,  the  e.m.f.  generated  in  the  secondary  circuit  is  se. 

Thus  the  secondary  current, 


where 

srl 


and  a2  = 


The  primary  exciting  current  is, 

loo  =  eY0  =  e  (g0  —  jbQ); 
thus,  the  total  primary  current, 

70  =  /i  +  Too  =  e  (bi  —  jb2), 
where, 

bi  =  fli  -f-  go,  and  62  —  #2  ~f-  bo. 

The  e.m.f.  consumed  by  the  primary  impedance  is, 
E1  =  I0Z0  =  e  (r0  +  jxQ)  (bi  -  j'62); 

the  primary  counter  e.m.f.  is  e,  thus  the  primary  impressed 
e.m.f., 


POLYPHASE  INDUCTION  MOTORS  233 

E0  =  e  +  El  =  6    ci  - 


where, 

d  =  1  +  r0&i  +  Xobz  and  02  = 

or,  the  absolute  value  is, 


eQ  =  e  Vci2  H-  c22, 
hence, 

=  e° 

Vci2  +  C22 

Substituting  this  value  gives, 
Secondary  current, 

/     _         «i  —  ja2 

\/Ci2  +  c22> 
Primary  current, 


Impressed  e.m.f., 

Ci 


VC!2+C22 

Thus  torque,  in  synchronous  watts  (that  is,  the  watts  output 
which  the  torque  would  produce  at  synchronous  speed), 

D  =  lehV 


hence,  the  torque  in  absolute  units, 


(Cl2  +  c22)  2wf 
where/  =  frequency. 

The  power  output  is  torque  times  speed,  thus: 


The  power  input  is, 

Po  =  [E  o/o]  =  [E  o/o]1  -  j 
e02  (bid 


234         ALTERNATING-CURRENT  PHENOMENA 
The  volt-ampere  input, 

Pan    =    £0/0    = 


+  622 


c2 


P(OWER  OUTPUT 
1000  2000  8000  4000  5000 


FIG.  123. 


hence,  the  efficiency  is, 


the  power-factor, 


POLYPHASE  INDUCTION  MOTORS 


the  apparent  efficiency, 
Pi 


the  torque  efficiency,1 


(1  -  s) 


235 


FIG.  124. 

and  the  apparent  torque  efficiency,2 
D  a 


C22) 

172.  Most  instructive  in  showing  the  behavior  of  an  induction 
motor  are  the  load  curves  and  the  speed  curves. 

The  load  curves  are  curves  giving,  with  the  power  output  as 
abscissas,  the  current  input,  speed,  torque,  power-factor,  effi- 
ciency, and  apparent  efficiency,  as.  ordinates. 

The  speed  curves  give,  with  the  speed  as  abscissas,  the  torque, 

1  That  is  the  ratio  of  actual  torque  to  torque  which  would  be  produced,  if 
there  were  no  losses  of  energy  in  the  motor,  at  the  same  power  input. 

2  That  is  the  ratio  of  actual  torque  to  torque  which  would  be  produced  if 
there  were  neither  losses  of  energy  nor  phase  displacement  in  the  motor,  at 
the  same  volt-ampere  input. 


236         ALTERNATING-CURRENT  PHENOMENA 

current  input,  power-factor,  torque  efficiency,  and  apparent 
torque  efficiency,  as  ordinates. 

The  load  curves  characterize  the  motor  especially  at  its 
normal  running  speeds  near  synchronism,  while  the  speed 
curves  characterize  it  over  the  whole  range  of  speed. 

In  Fig.  123  are  shown  the  load  curves,  and  in  Fig.  124  the 
speed  curves  of  a  motor  having  the  constants:  YQ  =  0.01  —  0.1  j; 
ZQ  =  0.1  +  0.3  j;  and  Zl  =  0.1  +  0.3  j. 


CHAPTER  XIX 
INDUCTION  GENERATORS 

173.  In  the  foregoing,  the  range  of  speed  from  s  =  1,  stand- 
still, to  s  =  0,  synchronism,  has  been  discussed.     In  this  range 
the  motor  does  mechanical  work. 

It  consumes  mechanical  power,  that  is,  acts  as  generator  or  as 
brake  outside  of  this  range. 

For  s  >  1,  backward  driving,  PI  becomes  negative,  repre- 
senting consumption  of  power,  while  D  remains  positive;  hence, 
since  the  direction  of  rotation  has  changed,  represents  con- 
sumption of  power  also.  All  this  power  is  consumed  in  the 
motor,  which  thus  acts  as  brake. 

For  s  <  0,  or  negative,  PI  and  D  become  negative,  and  the 
machine  becomes  an  electric  generator,  converting  mechanical 
into  electric  energy. 

The  calculation  of  the  induction  generator  at  constant  fre- 
quency, that  is,  at  a  speed  increasing  with  the  load  by  the 
negative  slip,  si,  is  the  same  as  that  of  the  induction  motor 
except  that  Si  has  negative  values,  and  the  load  curves  for  the 
machine  shown  as  motor  in  Fig.  122,  are  shown  in  Fig.  125  for 
negative  slip  Si  as  induction  generator. 

Again,  a  maximum  torque  point  and  a  maximum  output 
point  are  found,  and  the  torque  and  power  increase  from  zero 
at  synchronism  up  to  a  maximum  point,  and  then  decrease  again, 
while  the  current  constantly  increases. 

174.  The   induction    generator    differs   essentially   from    the 
ordinary   synchronous   alternator  in   so   far   as   the   induction 
generator  has  a  definite  power-factor,  while  the   synchronous 
alternator   has   not.     That  is,   in   the  synchronous   alternator 
the  phase  relation  between  current  and  terminal  voltage  entirely 
depends  upon  the  condition  of  the  external  circuit.     The  in- 
duction   generator,    however,    can    operate    only   if  the   phase 
relation  of  current  and  e.m.f.,  that  is,  the  power-factor  required 
by   the   external   circuit,    exactly   coincides   with   the   internal 
power-factor  of  the  induction  generator.     This  requires  that 

237 


238 


ALTERNATING-CURRENT  PHENOMENA 


the  power-factor  either  of  the  external  circuit  or  of  the  induction 
generator  varies  with  the  voltage,  so  as  to  permit  the  generator 
and  the  external  circuit  to  adjust  themselves  to  equality  of 
power-factor. 

Beyond  magnetic  saturation  the  power-factor  decreases; 
that  is,  the  lead  of  current  increases  in  the  induction  machine. 
Thus,  when  connected  to  an  external  circuit  of  constant  power- 
factor  the  induction  generator  will  either  not  generate  at  all, 
if  its  power-factor  is  lower  than  that  of  the  external  circuit,  or, 
if  its  power-factor  is  higher  than  that  of  the  external  circuit,  the 


ElECTBICAU   OUTPUT^  ,P 
-1000    I  -2000    |  -3000        -4000     (  -.soft)    |  -6000 


FIG.  125. 

voltage  will  rise  until  by  magnetic  saturation  in  the  induction 
generator  its  power-factor  has  fallen  to  equality  with  that  of 
the  external  circuit.  This,  however,  requires  magnetic  satura- 
tion in  the  induction  generator,  in  some  part  of  the  magnetic 
circuit,  as  for  instance  in  the  armature  teeth. 

To  operate  below  saturation — that  is,  at  constant  internal 
power-factor — the  induction  generator  requires  an  external 
circuit  with  leading  current,  whose  power-factor  varies  with  the 
voltage,  as  a  circuit  containing  synchronous  motors  or  syn- 
chronous converters.  In  such  a  circuit,  the  voltage  of  the 
induction  generator  remains  just  as  much  below  the  counter 
e.m.f.  of  the  synchronous  motor  as  is  necessary  to  give  the 


INDUCTION  GENERATORS  239 

required  leading  exciting  current  of  the  induction  generator,  and 
the  synchronous  motor  can  thus  to  a  certain  extent  be  called 
the  exciter  of  the  induction  generator. 

When  operating  self -exciting,  that  is,  shunt-wound,  con- 
verters from  the  induction  generator,  below  saturation  of  both 
the  converter  and  the  induction  generator,  the  conditions  are 
unstable  also,  and  the  voltage  of  one  of  the  two  machines  must 
rise  beyond  saturation  of  its  magnetic  field. 

When  operating  in  parallel  with  synchronous  alternating  cur- 
rent generators,  the  induction  generator  obviously  takes  its 
leading  exciting  current  from  the  synchronous  alternator,  which 
thus  carries  a  lagging  wattless  current. 

175.  To  generate  constant  frequency,  the  speed  of  the  in- 
duction generator  must  increase  with  the  load.  Inversely, 
when  driven  at  constant  speed,  with  increasing  load  on  the 
induction  generator,  the  frequency  of  the  current  generated 
thereby  decreases.  Thus,  when  calculating  the  characteristic 
curves  of  the  constant-speed  induction  generator,  due  regard 
has  to  be  taken  of  the  decrease  of  frequency  with  increase  of 
load,  or  what  may  be  called  the  slip  of  frequency,  s. 

Let,  in  an  induction  generator, 

YQ  =  00  —  j&o  =  primary  exciting  admittance, 

ZQ  =  ro  +  JXQ  =  primary  self-inductive  impedance, 

Zi  =  7*1  -\-jxi  =  secondary  self-inductive  impedance, 

reduced  to  primary,  all  these  quantities  being  reduced  to  the 
frequency  of  synchronism  with  the  speed  of  the  machine,  /. 

Let  e  =  generated  em.f.,  reduced  to  full  frequency. 

s  =  slip  of  frequency,  thus:  (1  —  s)  /  =  frequency  generated 
by  machine. 

We  then  have 
the  secondary  generated  e.m.f., 

se: 
thus,  the  secondary  current, 

.  1  ~  i 
where, 


, 
and  «2  = 


the  primary  exciting  current, 

7oo  =  EYQ  =  e  (gQ  -  jb0), 


240         ALTERNATING-CURRENT  PHENOMENA 
thus,  the  total  primary  current, 

70  =  /]  +  /oo  =  e(bi  -  jbz), 
where, 

bi  =  di  +  gQ  and  62  =  a2  +  &o; 

the  primary  impedance  voltage, 

El  =  70(r0+j[l  -  s]xb); 
the  primary  generated  e.m.f.  is, 

6(1    -«). 

Thus,  primary  terminal  voltage, 

E0  =  6(1  -  s)  -  70(r0  +  j[l  -  &]XQ)  =  e(ci  -  jc2), 


where, 

Ci  =  1  —  s  —  r06i  —  (1  —  s)zo&2  and  c2  =  (1  —  s)x0bi  —  r062, 
hence,  the  absolute  value  is, 


60   =  6\/Ci2  +  C22, 

and, 

=          e° 
=  Vci2  +  c22' 

Thus, 

the  secondary  current, 

1*  ~    °  /          ^T*  ^!  = 

the  primary  current, 

70  =    °    >    .    .  =^>  -^o  = 


the  primary  terminal  voltage, 

6p  (Ci  -  J 

: 


the  torque  and  mechanical  power  input, 


»  =  p-  =  i«f Jl  •  c-?r^ 


INDUCTION  GENERATORS 

the  electrical  output, 

Po  =  Po1  -  JPJ  =  [EoI0]  = 


-  j  (62ci  - 


the  volt-ampere  output, 


241 


2      » 


ELECTRICAL  OUTPUTrP  ,   WATTS 
aOQO          8000     I     4000          5QOO    I 


the  efficiency, 

•Pi 

the  power-factor, 


FIG.  126. 


+  &2C 


„        ^o1 

cos  6  =  TT-  — 


62c 


C22) 


or, 


tan  0 


_ 

Po1 


In  Fig.  126  is  plotted  the  load  characteristic  of  a  constant- 
speed  induction  generator,  at  constant  terminal  voltage  e0  =  110, 

16 


242         ALTERNATING-CURRENT  PHENOMENA 

and  the  constants:  F0  =  0.01  -  O.lj;  Z0  =  0.1  +  0.3  j,  and 
Zi  =  0.1  +  0.3  j. 

176.  As  an  example  may  be  considered  a  power  transmission 
from  an  induction  generator  of  constants  YQ}  Z0,  Zi,  over  a 
line  of  impedance,  Z  =  r  +  jz,  into  a  synchronous  motor  of 
synchronous  impedance,  Z2  =  r2  +  jz2,  operating  at  constant- 
field  excitation. 

Let  e0  —  counter  e.m.f.  or  nominal  generated  e.m.f.  of  syn- 
chronous motor  at  full  frequency;  that  is,  frequency  of  synchro- 
nism with  the  speed  of  the  induction  generator.  By  the  preced- 
ing paragraph  the  primary  current  of  the  induction  generator  was, 


Io  =  e(bi 
the  primary  terminal  voltage, 

EQ  =  e(ci  -  jc2)  ; 
thus,  terminal  voltage  at  synchronous  motor  terminals, 

E'0  =  E0  -  Jo  (r  +  j  [1  -  s]x) 

=  e(di  -  jd2), 
where, 

di  =  Ci  —  rbi  —  (1  —  s)  £&2  and  dz  =  c2  +  (1  —  s)  xbi  —  r&2; 
the  counter  e.m.f.  of  the  synchronous  motor, 

E2  =  Eo'  -  70  (r2  +  j  [1  -  s]x2) 

=  e(ki  -  jkz)  ; 
where, 

ki  =  di  —  r2bi  —  (1  —  s)  X2b2  and  kz  =  dz  +  (1  —  s)  £2&i  —  r2&2, 
or  the  absolute  value 


since,  however, 
we  have, 

Thus,  the  current, 

*•- 


60(1  —  s) 


INDUCTION  GENERATORS 

the  terminal  voltage  at  induction  generator, 
7?  _    go(l  -  s)  (ci  -  jc2) 


243 


OUTRUT  OF  SYNCHRONOUS,  WATTS 
8000       4000 


FIG.  127. 


and  the  terminal  voltage  at  the  synchronous  motor, 
,_eo(l  -  s)(di  -jd2). 


244         ALTERNATING-CURRENT  PHENOMENA 

herefrom  in  the  usual  way  the  efficiencies,  power-factor,  etc., 
are  derived. 

When  operated  from  an  induction  generator,  a  synchronous 
motor  gives  a  load  characteristic  very  similar  to  that  of  an 
induction  motor  operated  from  a  synchronous  generator,  but 
in  the  former  case  the  current  is  leading,  in  the  latter  lagging. 

In  either  case,  the  speed  gradually  falls  off  with  increasing 
load  (in  the  synchronous  motor,  due  to  the  falling  off  of  the 
frequency  of  the  induction  generator),  up  to  a  maximum  output 
point,  where  the  motor  drops  out  of  step  and  comes  to  standstill. 

Such  a  load  characteristic  of  the  induction  generator  in  Fig. 
126,  feeding  a  synchronous  motor  of  counter  e.m.f.  e0  =  125  volts 
(at  full  frequency)  and  synchronous  impedance  Z2  =  0.04  +  6  j, 
over  a  line  of  negligible  impedance  is  shown  in  Fig.  127. 


CHAPTER  XX 
SINGLE-PHASE  INDUCTION  MOTORS 

177.  The  magnetic  circuit  of  the  induction  motor  at  or  near 
synchronism  consists  of  two  magnetic  fluxes  superimposed  upon 
each  other  in  quadrature,  in  time,  and  in  position.  In  the 
polyphase  motor  these  fluxes  are  produced  by  e.m.fs.  displaced 
in  phase.  In  the  monocyclic  motor  one  of  the  fluxes  is  due  to 
the  primary  power  circuit,  the  other  to  the  primary  exciting 
circuit.  In  the  single-phase  motor  the  one  flux  is  produced  by 
the  primary  circuit,  the  other  by  the  currents  produced  in  the 
secondary  or  armature,  which  are  carried  into  quadrature  posi- 
tion by  the  rotation  of  the  armature.  In  consequence  thereof, 
while  in  all  these  motors  the  magnetic  distribution  is  the  same 
at  or  near  synchronism,  and  can  be  represented  by  a  rotating 
field  of  uniform  intensity  and  uniform  velocity,  it  remains  such 
in  polyphase  and  monocyclic  motors;  but  in  the  single-phase 
motor,  with  increasing  slip — that  is,  decreasing  speed — the  - 
quadrature  field  decreases,  since  the  secondary  armature  cur- 
rents are  not  carried  to  complete  quadrature  position;  and  thus 
only  a  component  is  available  for  producing  the  quadrature  flux. 
Hence,  approximately,  the  quadrature  flux  of  a  single-phase 
motor  can  be  considered  as  proportional  to  its  speed;  that  is, 
it  is  zero  at  standstill. 

Since  the  torque  of  the  motor  is  proportional  to  the  product 
of  secondary  current  times  magnetic  flux  in  quadrature,  it 
follows  that  the  torque  of  the  single-phase  motor  is  equal  to 
that  of  the  same  motor  under  the  same  condition  of  operation 
on  a  polyphase  circuit,  multiplied  with  the  speed;  hence  equal 
to  zero  at  standstill. 

Thus,  while  single-phase  induction  motors  are  quite  satisfac- 
tory at  or  near  synchronism,  their  torque  decreases  proportionally 
with  the  speed,  and  becomes  zero  at  standstill.  That  is,  they 
are  not  self -starting,  but  some  starting  device  has  to  be  used. 

Such  a  starting  device  may  either  be  mechanical  or  electrical. 
All  the  electrical  starting  devices  essentially  consist  in  impress- 

245 


246         ALTERNATING-CURRENT  PHENOMENA 

ing  upon  the  motor  at  standstill  a  magnetic  quadrature  flux. 
This  may  be  produced  either  by  some  outside  e.m.f.,  as  in  the 
monocyclic  starting  device,  or  by  displacing  the  circuits  of  two 
or  more  primary  coils  from  each  other,  either  by  mutual  induc- 
tion between  the  coils — that  is,  by  using  one  as  secondary 
to  the  other — or  by  impedances  of  different  inductance  factors 
connected  with  the  different  primary  coils. 

178.  The  starting  devices  of  the  single-phase  induction  motor 
by  producing  a  quadrature  magnetic  flux  can  be  subdivided 
into  three  classes: 

1.  Phase-Splitting  Devices.     Two  or  more  primary  circuits 
are  used,  displaced  in  position  from  each  other,  and  either  in 
series  or  in  shunt  with  each  other,  or  in  any  other  way  related, 
as  by  transformation.     The  impedances  of  these  circuits  are 
made  different  from  each  other  as  much  as  possible  to  produce 
a  phase  displacement  between  them.     This  can  be  done  either 
by  inserting  external  impedances  in  the  circuits,  as  a  condenser 
and  a  reactive  coil,  or  by  making  the  internal  impedances  of  the 
motor  circuits  different,  as  by  making  one  coil  of  high  and  the 
other  of  low  resistance. 

2.  Inductive    Devices.     The    different    primary    circuits    of 
the  motor  are  inductively  related  to  each  other  in  such  a  way 
as  to  produce  a  phase  displacement  between  them.     The  induct- 
ive relation  can  be  outside  of  the  motor  or  inside,  by  having 
the  one  coil  submitted  to  the  inductive  action  of  the  other;  and 
in  this  latter  case  the  current  in  the  secondary  coil  may  be  made 
leading,  accelerating  coil,  or  lagging,  shading  coil. 

3.  Monocyclic  Devices.     External  to  the  motor  an  essentially 
wattless  e.m.f.  is  produced  in  quadrature  with  the  main  e.m.f. 
and  impressed  upon  the  motor,  either  directly  or  after  com- 
bination   with    the    single-phase    main    e.m.f.     Such    wattless 
quadrature  e.m.f.  can  be  produced  by  the  common  connection 
of  two  impedances   of  different  power-factor,  as  an  inductive 
reactance  and  a  resistance,  or  an  inductive  and  a  condensive 
reactance  connected  in  series  across  the  mains. 

The  investigation  of  these  starting-devices  offers  a  very 
instructive  application  of  the  symbolic  method  of  investiga- 
tion of  alternating-current  phenomena,  and  a  study  thereof 
is  thus  recommended  to  the  reader.1 

1  See  paper  on  the  Single-phase  Induction  Motor,  A.  I.  E.  E.  Transactions, 
1898. 


SINGLE-PHASE  INDUCTION  MOTORS  247 

179.  Occasionally,  no  special  motors  are  built  for  single-phase 
operation,  but  polyphase  motors  used  in  single-phase  circuits, 
since  for  starting  the  polyphase  primary  winding  is  required, 
the  single  primary-coil  motor  obviously  not  allowing  the  appli- 
cation of  phase-displacing  devices  for  producing  the  starting 
quadrature  flux. 

Since  at  or  near  synchronism,  at  the  same  impressed  e.m.f. 
— that  is,  the  same  magnetic  density — the  total  volt-amperes 
excitation  of  the  single-phase  induction  motor  must  be  the  same 
as  of  the  same  motor  on  polyphase  circuit,  it  follows  that  by 
operating  a  quarter-phase  motor  from  single-phase  circuit  on 
one  primary  coil,  its  primary  exciting  admittance  is  doubled. 
Operating  a  three-phase  motor  single-phase  on  one  circuit  its 
primary  exciting  admittance  is  trebled.  The  self-inductive 
primary  impedance  is  the  same  single-phase  as  polyphase,  but 
the  secondary  impedance  reduced  to  the  primary  is  lowered, 
since  in  single-phase  operation  all  secondary  circuits  corre- 
spond to  the  one  primary  circuit  used.  Thus  the  secondary 
impedance  in  a  quarter-phase  motor  running  single-phase  is 
reduced  to  one-half,  in  a  three-phase  motor  running  single- 
phase  reduced  to  one-third.  In  consequence  thereof  the  slip  of 
speed  in  a  single-phase  induction  motor  is  usually  less  than  in  a 
polyphase  motor;  but  the  exciting  current  is  considerably 
greater,  and  thus  the  power-factor  and  the  efficiency  are  lower. 

The  preceding  considerations  obviously  apply  only  when 
running  so  near  synchronism  that  the  magnetic  field  of  the 
single-phase  motor  can  be  assumed  as  uniform,  that  is,  the 
cross-magnetizing  flux  produced  by  the  armature  as  equal  to 
the  main  magnetic  flux. 

When  investigating  the  action  of  the  single-phase  motor  at 
lower  speeds  and  at  standstill,  the  falling  off  of  the  magnetic 
quadrature  flux  produced  by  the  armature  current,  the  change 
of  secondary  impedance,  and  where  a  starting  device  is  used 
the  effect  of  the  magnetic  field  produced  by  tne  starting  device, 
have  to  be  considered. 

The  exciting  current  of  the  single-phase  motor  consists  of 
the  primary  exciting  current  or  current  producing  the  main 
magnetic  flux,  and  represented  by  a  constant  admittance,  Fo1, 
the  primary  exciting  admittance  of  the  motor,  and  the  secondary 
exciting  current,  that  is,  that  component  of  primary  current 
corresponding  to  the  secondary  current  which  gives  the  excita- 


248         ALTERNATING-CURRENT  PHENOMENA 

tion  for  the  quadrature  magnetic  flux.  This  latter  magnetic 
flux  is  equal  to  the  main  magnetic  flux,  $o,  at  synchronism, 
and  falls  off  with  decreasing  speed  to  zero  at  standstill,  if  no 
starting  device  is  used,  or  to  3>i  =  $0£  at  standstill  if  by  a  start- 
ing device  a  quadrature  magnetic  flux  is  impressed  upon  the 
motor,  and  at  standstill  t  =  ratio  of  quadrature  or  starting 
magnetic  flux  to  main  magnetic  flux. 

Thus  the  secondary  exciting  current  can  be  represented  by  an 
admittance,  IV,  which  changes  from  equality  with  the  primary 
exciting  admittance,  IV  at  synchronism  to  Fi1  =  0,  respect- 
ively to  Fi1  =  ZFo1  at  standstill.  Assuming  thus  that  the 
starting  device  is  such  that  its  action  is  not  impaired  by  the 
change  of  speed,  at  slip  s  the  secondary  exciting  admittance 
can  be  represented  by:  . 

Fi1  =  [1  -  (1  -  0  s]  Fo1. 

The  secondary  impedance  of  the  motor  at  synchronism  is 
the  joint  impedance  of  all  the  secondary  circuits,  since  all 

secondary    circuits   correspond    to   the   same   primary   circuit, 

7  7 

hence  =  -5-   with   a  three-phase  secondary,   and  =  -*•  with  a 

two-phase  secondary  with  impedance  Z\  per  circuit. 

At  standstill,  however,  the  secondary  circuits  correspond  to 
the  primary  circuit  only  with  their  projection  in  the  direction 
of  the  primary  flux,  and  thus  as  resultant  only  one-half  of  the 
secondary  circuits  are  effective,  so  that  the  secondary  impe- 

2  7 

dance  at  standstill  is  equal  to  — -^  with  a  three-phase,  and  equal 

o 

to  Zi  with  a  two-phase,  secondary.  Thus  the  effective  second- 
ary impedance  of  the  single-phase  motor  changes  with  the  speed 

and  can  at  the  slip  s  be  represented  by  Zi1  =  Q -1  in    a 

o 

three-phase   secondary,    and    Zi1  =  ~ •    in   a    two-phase 

z 

secondary,  with  the  impedance  Zi  per  secondary  circuit. 

In  the  single-phase  motor  without  starting  device,  due  to 
the  falling  off  of  the  quadrature  flux,  the  torque  at  slip  s  is : 

D  =  a^  (1  -  s). 

(a  and  e  see  paragraph  171.) 

In   a   single-phase   motor   with   a   starting   device   which   at 


SINGLE-PHASE  INDUCTION  MOTORS  249 

standstill  produces  a  ratio  of  magnetic  fluxes  t,  the  torque  at 
standstill  is 

Do  =  tDi, 

where  DI  =  a\e^  =  total  torque  of  the  same  motor  on  polyphase 
circuit. 

Thus  denoting  the  value  -^~  =  v,  the  single-phase  motor  torque 

at  standstill  is: 

Z)0  =vDi=  aie2v, 

and  the  single-phase  motor  torque  at  slip  s  is : 
D  =  aie*[l  -  (I  -  v)  s]. 

180.  In  the  single-phase  motor  considerably  more  advan- 
tage is  gained  by  compensating  for  the  wattless  magnetizing 
component  of  current  by  capacity  than  in  the  polyphase  motor, 
where  this  wattless  component  of  the  current  is  relatively 
small.  The  use  of  shunted  capacity,  however,  has  the  dis- 
advantage of  requiring  a  wave  of  impressed  e.m.f.  very  close 
to  sine  shape,  since  even  with  a  moderate  variation  from  sine 
shape  the  wattless  charging  current  of  the  condenser  of  higher 
frequency  may  lower  the  power-factor  more  than  the  compen- 
sation for  the  wattless  component  of  the  fundamental  wave 
raises  it,  as  will  be  seen  in  the  chapter  on  General  Alternating- 
current  Waves. 

Thus  the  most  satisfactory  application  of  the  condenser 
in  the  single-phase  motor  is  not  in  shunt  to  the  primary  circuit, 
but  in  a  tertiary  circuit;  that  is,  in  a  circuit  stationary  with 
regard  to  the  primary  impressed  circuit  but  submitted  to  in- 
ductive action  by  the  revolving  secondary  circuit. 

In  this  case  the  condenser  is  supplied  with  an  e.m.f.  trans- 
formed twice,  from  primary  to  secondary  and  from  secondary 
to  tertiary,  through  multitooth  structures  in  a  uniformly  re- 
volving field,  and  thus  a  very  close  approximation  to  sine  wave 
produced  at  the  condenser,  irrespective  of  the  wave-shape 
of  primary  impressed  e.m.f. 

With  the  condenser  connected  into  a  tertiary  circuit  of  a 
single-phase  induction  motor,  the  wattless  magnetizing  current 
of  the  motor  is  supplied  by  the  condenser  in  a  separate  circuit, 
and  the  primary  coil  carries  the  power  current  only,  and  thus 
the  efficiency  of  the  motor  is  essentially  increased. 


250         ALTERNATING-CURRENT  PHENOMENA 

The  tertiary  circuit  may  be  at  right  angles  to  the  primary, 
or  under  any  other  angle.  Usually  it  is  applied  on  an  angle 
of  45°  to  60°,  so  as  to  secure  a  mutual  induction  between  tertiary 
and  primary  for  starting,  which  produces  in  starting  in  the  con- 
denser a  leading  current,  and  gives  the  quadrature  magnetic 
flux  required. 

181.  The  most  convenient  way  to  secure  this  arrangement 
is  the  use  of  a  three-phase  motor  which  with  two  of  its  ter- 
minals, 1-2,  is  connected  to  the  single-phase  mains,  and  with 
terminals  1  and  3  to  a  condenser. 

Let  FO  =  go  —  jbQ  =  primary  excitin'g  admittance  of  the  motor 
per  delta  circuit. 

ZQ  =  r0  +  jxo  =  primary  self-inductive  impedance  per  delta 
circuit. 

Zi  =  7*1  +  jXi  =  secondary  self-inductive  impedance  per  delta 
circuit  reduced  to  primary. 

Let 

F3  =  03  +  jb3  =  admittance  of  the  condenser  connected  be- 
tween terminals  1  and  3. 

If  then,  as  single-phase  motor, 

t  —  ratio  of  auxiliary  quadrature  flux  to  main  flux  in 

starting, 
h  =  ratio  of  e.m.f.  generated  in   condenser  circuit  to 

e.m.f.  generated  in  main  circuit  in  starting, 
starting  torque 
aie2  in  starting 

Operating  single-phase 

3V  =  1.5  F0  =  1.5(00  —  jbo)  =  primary    exciting    admit- 
tance; 

Fi1  =  1.5  F0[l  -  (1  -  0  s] 

=  1.5  (gQ  —  jbo)  [1  —  (1  —  t)  s]  =  secondary  exciting 
admittance  at  slip  s; 

2Z0      2(r0  +  jxo) 

Zo1  =  —5-  =  — - — gr^    -  =  primary  self-inductive  impe- 
o  6 

dance; 

Zli  =  (L+_!)  Zl  =  (1+j)  (ri  +  jsxj  =  secondary    self- 
inductive  impedance; 

Z2i  =  2Z°  =  2(r0+jx0)  =  tertiary  self.inductive  impe- 

o  o 

dance  of  motor. 


SINGLE-PHASE  INDUCTION  MOTORS  251 

Thus, 

F4  =  -        —     =  total  admittance  of  tertiary  circuit. 


Since  the  e.m.f.  generated  in  the  tertiary  circuit  decreases 
from  e  at  synchronism  to  he  at  standstill,  the  effective  tertiary 
admittance  or  admittance  reduced  to  a  generated  e.m.f.,  e,  is 
at  slip  s, 

F4L  =  [1  -  (1  -  K)s]Yt. 
Let  then, 

e  =  counter  e.m.f.  of  primary  circuit, 
s  =  slip. 
We  have, 
the  secondary  load  current, 

=  €(a'  -  ja^' 


the  secondary  exciting  current, 

/ji  =  eY^  =  1.5  eYo  [1  -  (1  -  t)  [s; 
the  secondary  condenser  current; 

thus,  the  total  secondary  current, 

the  primary  exciting  current, 
V  =  eYQl  =  1.5  eY0, 
thus,  the  total  primary  current, 

70  =  71  +  /o1  =  /i  +  /4  +  7!1  +  7e?  =  e(b,  -  J62); 
the  primary  impressed  e.m.f., 

EQ  =  e  +  ZQ1I0  =  e(ci  —  jc2); 
thus,  the  main  counter  e.m.f., 

e  =  — _°.    t 
or, 


252         ALTERNATING-CURRENT  PHENOMENA 
and  the  absolute  value, 


eo 

e  = 


hence,  the  primary  current, 

T        eQ(b1  -  j 
•to  =  ^ 


or, 

The  volt-ampere  input, 

the  power  input, 

p   =|7eli=     2MijH>2C2. 

Ci2  +  C22   ' 

the  torque  at  slip  s, 

and  the  power  output, 

P  =  D  (1  -  s) 


and  herefrom  in  the  usual  manner  may  be  derived  the  efficiency, 
apparent  efficiency,  torque  efficiency,  apparent  torque  efficiency, 
and  power-factor. 

The  derivation  of  the  constants,  t,  h,  v,  which  have  to  be 
determined  before  calculating  the  motor,  is  as  follows: 

Let  eQ  =  single-phase  impressed  e.m.f., 

Y  =  total  stationary  admittance  of  motor  per  delta  circuit, 
EZ  =  e.m.f.  at  condenser  terminals  in  starting. 

In  the  circuit  between  the  single-phase  mains  from  terminal 
1  over  terminal  3  to  2,  the  admittances,  Y  +  Fa,  and  F,  are  con- 
nected in  series,  and  have  the  respective  e.m.f  s.,  E%  and  eQ  —  E3. 
It  is  thus, 

Trr      i       T/-       _._     T/1    777       _._     77F 

since  with  the  same  current  in  both   circuits,   the  impressed 

e.m.fs.  are  inversely  proportional  to  the  respective  admittances. 

Thus, 

F  e°F 

^  ~2  Y  +  F3  = 


SINGLE-PHASE  INDUCTION  MOTORS 
and  the  quadrature  e.m.f.  is 

hence, 
and 


253 


+  7i22. 

Since  in  the  three-phase  e.m.f.  triangle,  the  altitude  corre- 
sponding  to   the    quadrature   magnetic    flux  =  — ^=»  and  the 

•""v  3 

quadrature  and  main  fluxes  are  equal,  in  the  single-phase  motor 
the  ratio  of  quadrature  to  main  flux  is 

t  =  ~  =  1.1557*2. 

v  3 

From  t,  v  is  derived  as  shown  in  the  preceding. 
182.  The  most  frequently  used  starting  device  of  single-phase 
induction  motors  (with  the  exception  of  fan  motors,  in  which  the 


E.I.Y, 


|*,Jf,Y| 


FIG.  128. 

shading  coil  is  commonly  used)  is  the  monocyclic  starting  device. 
It  consists  in  producing  externally  to  the  motor  a  system  of 
polyphase  e.m.fs.  with  single-phase  flow  of  energy,  and  im- 
pressing it  upon  the  motor,  which  is  wound  as  polyphase,  usually 
three-phase  motor. 

Such  a  polyphase  system  of  e.m.fs.  with  single-phase  flow  of 
energy  has  been  called  a  monocyclic  system.  It  essentially 
consists,  or  can  be  resolved  into,  a  main  or  energy  e.m.f,,  in 
phase  with  the  flow  of  energy,  and  an  auxiliary  or  wattless  e.m.f. 
in  quadrature  thereto. 

If  across  the  single-phase  mains  of  voltage,  e,  two  impedances 
of  different  inductance  factors,  of  the  respective  admittances, 
Yi  and  F2,  are  connected,  the  voltages,  EI  and  E2  of  these  im- 


254         ALTERNATING-CURRENT  PHENOMENA 

pedances  are  displaced  from  each  other,  thus  forming  with  the 
main  voltage,  e,  a  voltage  triangle,  or  a  more  or  less  distorted 
three-phase  system,  as  shown  in  Fig.  128. 

Connecting  now  a  three-phase  induction  motor  with  two  of 
its  terminals,  1  and  2,  to  the  single-phase  mains  a,  and  6,  and 
with  its  third  terminal  3  to  the  common  connection,  c,  of  the  two 
impedances,  a  quadrature  flux  is  produced  in  this  motor,  by  the 
traverse  voltage,  ES}  of  the  monocyclic  triangle,  Fig.  128. 

It  is  then: 

#1  +  E2  =  e  (1) 

EZ  —  EI  =  ES  (2) 

hence: 


e 

?2  ~2 
Let  now,  in  Fig.  128. 
Y  =  effective  admittance  of  motor  between  terminals  1  and 

2  at  standstill. 

Y3  =  effective  admittance  of  motor  for  the  quadrature  flux, 
from  terminal  3  to  middle  between  1  and  2. 


As  the  voltage  of  this  latter  admittance  is  -^-\/3^  the  altitude 

z 

of  the  three-phase  motor  triangle,  and  as  the  magnetic  flux  is  the 
same  in  all  directions,  in  the  polyphase  motor,  and  the  effective 
admittances  are  proportional  to  the  square  of  the  voltage,  it  is: 


Y,  +  y  =  « 


g 


hence: 

Y3  =  |F 

Denoting  the  currents  and  voltages  in  the  direction  as  shown 
by  the  arrows  in  Fig.  128,  it  is: 

7,    =    /!-/,  (4) 

and: 

73  =  F3#3  =  |  YE,  (5) 


SINGLE-PHASE  INDUCTION  MOTORS  255 

\  =  YlE1  =  Yife  - 

(6) 


(By  equation  (3))  substituting  (5)  and  (6)  into  (4),  gives,  after 
transposing: 

e         Y,  -  F2 

•         2    ._     L  v    L4      •  (7) 


Substituting  (7)  into  (3),  (5),  (6)  then  gives  the  voltages  and 
currents : 

Ei,  EZ,  /a,  Iij  Iz 

The  current  traversing  the  motor  from  terminal  1  to  terminal  2 
is 

If  =  eY  (8) 

and  upon  this  superimpose  the  return  of  the  current  /3,  so  that 
current 

I'*    =   6F  +  ^/3  (9) 

leaves  terminal  2,  and  current 

f'i  =  eY  -  ~  73  (10) 

enters  terminal  1. 

The  total  current  taken  by  the  motor  and  starting  device  from 
the  single-phase  mains  then  is: 

/  =  h  +  /'i    1 

(ID 


and  herefrom  follows  the  volt-ampere  input: 

Q  =  el  (12) 

while  on  polyphase  supply,  the  volt-ampere  input  is: 

QQ  =2  el'  =  2e2Y  (13) 

thus  the  ratio  of  volt-ampere  inputs  is: 

Q        I 


Qo       2eY 


(14) 


The  ratio  of  the  starting  torque  of  the  motor  with  the  monocyc- 
lic  starting  device,  to  that  of  the  same  motor  on  three-phase 


256         ALTERNATING-CURRENT  PHENOMENA 

supply,  is  the  ratio  of  the  quadrature  fluxes,  which  is  proportional 
to  the  quadrature  voltages: 

BJ        j  y.  -  ' 

== 


where  the  index,  jy  denotes,  that  only  the  quadrature  term  of  the 
expression  is  effective  in  producing  torque. 

The  ratio  of  the  apparent  starting  torque  efficiencies  thus  is  : 

(16) 

183.  Usually  a  resistance  and  a  reactance  are  used  as  the  two 
impedances  of  the  monocyclic  starting  device,  as  the  cheapest, 
though  the  triangle  produced  thereby  has  a  low  altitude,  E2)  and 
starting  torque  and  torque  efficiency  thus  are  comparatively  low. 

Let  as  illustration,  in  the  three-phase  motor,  Figs.  122  and  123, 
a  resistance-reactance  starting  device  be  used  of  the  values  :  r  = 
1  ohm,  and  x  =  1  ohm  hence: 


In  this  motor,  at  standstill,  it  is,  per  delta  circuit: 

(a)  Without  start-     (6)  With  secondary 
ing  resistance  :  resistance    i  n  - 

creased  ten  fold: 

Voltage:  e  =  110  volts 

Current:  i  =  176  amp.  8.97  amp. 

Torque:  D  =  2.93  syn.  kw.        7.38  syn.  kw. 

Power-factor:  p  =  0.313  0.835 

Hence  the  current, 

vectorially:  /  =  55  -  167  j  75  -  49  j 

and  the  admittance,  per  motor 

circuit:  Y'  =  0.5  -  1.52  j       0.68  -  0.45  j 

Hence,  the  effective  admittance,  between  two  motor  terminals  1 
and  2: 

Y  =  1.5  Y'  =  0.75  -  2.28  j     1.02  -  0.67  j 

Herefrom  follows: 

Quadrature  voltage:          E9  =  -  5.5  +  16.3  j     2.7  +  25.5  j 


SINGLE-PHASE  INDUCTION  MOTORS  257 

Relative    starting 

torque:  t  =  0.172  0.268 

Starting  torque:  3  tD  =  1.52  syn.  kw.  6.73  syn.  kw. 

As  seen,  with  starting  resistance  in  the  secondary  circuit,  a 
fairly  good  starting  torque  is  given  by  this  device;  but  with 
short-circuited  armature,  the  starting  torque  is  low. 

184.  The  greater  the  difference  in  the  inductance  factors  of 
the  two  impedances  in  the  starting  device,  the  higher  values  of 
quadrature  voltage,  E3,  and  thus  of  starting  torque  are  available. 

The  combination  of  inductance  and  capacity  thus  gives  the 
highest  torque,  and  by  such  combination,  true  three-phase  rela- 
tion can  be  secured,  that  is,  the  conditions  brought  about: 

El  =  E2  =  e 

The  starting  by  condenser  in  the  tertiary  circuit,  of  a  three- 
phase  motor,  can  be  considered  as  a  special  case  of  the  mono- 
cyclic  starting  device,  for  FI  =  0  and  F2  =  capacity  susceptance. 

A  further  extension  of  the  monocyclic  starting  device  is,  to  use 
another  induction  motor,  which  is  running  at  speed,  to  supply 
the  quadrature  voltage,  Es. 

Thus,  if  a  number  of  single-phase  induction  motors  are  oper- 
ated near  each  other,  as  in  the  same  factory,  etc.,  they  can  all  be 
made  self-starting — except  the  first  one — by  connecting  their 
third  terminals  together.  That  is,  connecting  a  number  of  three- 
phase  induction  motors,  with  two  of  their  terminals,  1,  2  to 
single-phase  mains  a,  6,  and  connecting  all  their  third  terminals,  3, 
with  each  other  by  an  interconnecting  main,  c,  then,  as  soon  as 
one  of  the  motors  is  running,  all  the  others  can  be  started  by 
drawing  quadrature  voltage  and  current  from  the  one  which  is 
running. 

This  is  a  convenient  means  of  operating  single-phase  induction 
motors  self-starting  without  separate  starting  devices.  It  has 
the  further  advantage,  that  an  overloaded  motor  begins  to  draw 
current  over  the  interconnecting  circuit,  c,  from  the  other  motors, 
as  phase  converters,  and  the  maximum  output  of  the  individual 
motors  thereby  is  increased  far  beyond  that  of  the  motor  as 
single-phase  motor,  near  to  that  as  three-phase  motor. 

As  single-phase  motors,  especially  with  armature  resistance, 
when  once  started  and  when  not  loaded,  speed  up  from  low  speed 

17 


258         ALTERNATING-CURRENT  PHENOMENA 

to  full  speed,  the  first  motor  in  such  monocyclic  interconnecting 
system  can  be  started  by  hand,  after  taking  its  load  off. 

For  further  discussion  on  the  theory  and  calculation  of  the 
single-phase  induction  motor,  see  American  Institute  Electrical 
Engineers  Transactions,  January,  1898  and  1900. 


SECTION  V 
SYNCHRONOUS  MACHINES 


CHAPTER  XXI 
ALTERNATING-CURRENT  GENERATOR 

185.  In  the  alternating-current  generator,  e.m.f.  is  generated 
in  the  armature  conductors  by  their  relative  motion  through  a 
constant  or  approximately  constant  magnetic  field. 

When  yielding  current,  two  distinctly  different  m.m.fs.  are 
acting  upon  the  alternator  armature — the  m.m.f.  of  the  field 
due  to  the  field-exciting  spools,  and  the  m.m.f.  of  the  armature 
current.  The  former  is  constant,  'or  approximately  so,  while  the 
latter  is  alternating,  and  in  synchronous  motion  relatively  to  the 
former;  hence  fixed  in  space  relative  to  the  field  m.m.f.,  or  uni- 


FIG.  129. 

directional,  but  pulsating  in  a  single-phase  alternator.  In  the 
polyphase  alternator,  when  evenly  loaded  or  balanced,  the  result- 
ant m.m.f.  of  the  armature  current  is  more  or  less  constant. 

The  e.m.f.  generated  in  the  armature  is  due  to  the  magnetic 
flux  passing  through  and  interlinked  with  the  armature  con- 
ductors. This  flux  is  produced  by  the  resultant  of  both  m.m.fs., 
that  of  the  field,  and  that  of  the  armature. 

On  open-circuit,  the  m.m.f.  of  the  armature  is  zero,  and  the 
e.m.f.  of  the  armature  is  due  to  the  m.m.f.  of  the  field-coils  only. 
In  this  case  the  e.m.f.  is,  in  general,  a  maximum  at  the  moment 
when  the  armature  coil  faces  the  position  midway  between 
adjacent  field-coils,  as  shown  in  Fig.  129,  and  thus  incloses 

259 


260         ALTERNATING-CURRENT  PHENOMENA 

no  magnetism.     The  e.m.f.   wave  in   this   case  is,  in  general, 
symmetrical. 

An  exception  to  this  statement  may  take  place  only  in  those 
types  of  alternators  where  the  magnetic  reluctance  of  the  arma- 
ture is  different  in  different  directions;  thereby,  during  the  syn- 
chronous rotation  of  the  armature,  a  pulsation  of  the  magnetic 
flux  passing  through  it  is  produced.  This  pulsation  of  the  mag- 
netic flux  generates  e.m.f.  in  the  field-spools,  and  thereby  makes 
the  field  current  pulsating  also.  Thus,  we  have,  in  this  case,  even 
on  open-circuit,  no  rotation  through  a  constant  magnetic  field, 
but  rotation  through  a  pulsating  field,  which  makes  the  e.m.f. 
wave  unsymmetrical,  and  shifts  the  maximum  point  from  its 
theoretical  position  midway  between  the  field-poles.  In  general 
this  secondary  reaction  can  be  neglected,  and  the  field  m.m.f. 
be  assumed  as  constant. 


FIG.  130. 

The  relative  position  of  the  armature  m.m.f.  with  respect  to 
the  field  m.m.f.  depends  upon  the  phase  relation  existing  in  the 
electric  circuit.  Thus,  if  there  is  no  displacement  of  phase  be- 
tween current  and  e.m.f.,  the  current  reaches  its  maximum  at 
the  same  moment  as  the  e.m.f.  or,  in  the  position  of  the  armature 
shown  in  Fig.  129,  midway  between  the  field-poles.  In  this  case 
the  armature  current  tends  neither  to  magnetize  nor  demagnetize 
the  field,  but  merely  distorts  it;  that  is,  demagnetizes  the  trail- 
ing pole  corner,  a,  and  magnetizes  the  leading  pole  corner,  b. 
A  change  of  the  total  flux,  and  thereby  of  the  resultant  e.m.f., 
will  take  place  in  this  case  only  when  the  magnetic  densities  are 
so  near  to  saturation  that  the  rise  of  density  at  the  leading  pole 
corner  will  be  less  than  the  decrease  of  density  at  the  trailing 
pole  corner.  Since  the  internal  self-inductive  reactance  of  the 
alternator  itself  causes  a  certain  lag  of  the  current  behind  the 
generated  e.m.f.,  this  condition  of  no  displacement  can  exist  only 
in  a  circuit  with  external  negative  reactance,  as  capacity,  etc. 


ALTERNATING-CURRENT  GENERATOR          261 

If  the  armature  current  lags,  it  reaches  the  maximum  later 
than  the  e.m.f. ;  that  is,  in  a  position  where  the  armature-coil 
partly  faces  the  field-pole  which  it  approaches,  as  shown  in  dia- 
gram in  Fig.  130.  Since  the  armature  current  is  in  ppposite  direc- 
tion to  the  current  in  the  following  field-pole  (in  a  generator),  the 
armature  in  this  case  will  tend  to  demagnetize  the  field. 

If,  however,  the  armature  current  leads — that  is,  reaches  its 
maximum  while  the  armature-coil  still  partly  faces  the  field-pole 
which  it  leaves,  as  shown  in  diagram,  Fig.  131 — it  tends  to 
magnetize  this  field-pole,  since  the  armature  current  is  in  the 
same  direction  as  the  exciting  current  of  the  preceding  field 
spools. 

Thus,  with  a  leading  current,  the  armature  reaction  of  the 
alternator  strengthens  the  field,  and  thereby,  at  constant  field 
excitation,  increases  the  voltage;  with  lagging  current  it  weakens 


FIG.  131. 

the  field,  and  thereby  decreases  the  voltage  in  a  generator.  Ob- 
viously, the  opposite  holds  for  a  synchronous  motor,  in  which  the 
armature  current  is  in  the  opposite  direction;  and  thus  a  lagging 
current  tends  to  magnetize,  a  leading  current  to  demagnetize, 
the  field. 

186.  The  e.m.f.  generated  in  the  armature  by  the  resultant 
magnetic  flux,  produced  by  the  resultant  m.m.f.  of  the  field  and 
of  the  armature,  is  not  the  terminal  voltage  of  the  machine;  the 
terminal  voltage  is  the  resultant  of  this  generated  e.m.f.  and  the 
e.m.f.  of  self-inductive  reactance  and  the  e.m.f.  representing  the 
power  loss  by  resistance  in  the  alternator  armature.  That  is, 
in  other  words,  the  armature  current  not  only  opposes  or  assists 
the  field  m.m.f.  in  creating  the  resultant  magnetic  flux,  but  sends 
a  second  magnetic  flux  in  a  local  circuit  through  the  armature, 
which  flux  does  not  pass  through  the  field-spools,  and  is  called 
the  magnetic  flux  of  armature  self-inductive  reactance. 


262         AL  TERN  A  TING-C  URREN  T  PHENOMENA 

Thus  we  have  to  distinguish  in  an  alternator  between  armature 
reaction,  or  the  magnetizing  action  of  the  armature  upon  the 
field,  and  armature  self-inductive  reactance,  or  the  e.m.f.  gener- 
ated in  the  armature  conductors  by  the  current  therein.  This 
e.m.f.  of  self-inductive  reactance  is  (if  the  magnetic  reluctance, 
and  consequently  the  reactance,  of  the  armature  circuit  is  as- 
sumed as  constant)  in  quadrature  behind  the  armature  current, 
and  will  thus  combine  with  the  generated  e.m.f.  in  the  proper 
phase  relation.  Obviously  the  e.m.f.  of  self-inductive  reactance 
and  the  generated  e.m.f.  do  not  in  reality  combine,  but  their 
respective  magnetic  fluxes  combine  in  the  armature-core,  where 
they  pass  through  the  same  structure.  These  component  e.m.fs. 
are  therefore  mathematical  fictions,  but  their  resultant  is  real. 
This  means  that,  if  the  armature  current  lags,  the  e.m.f.  of  self- 
inductive  reactance  will  be  more  than  90°  behind  the  generated 
e.m.f.,  and  therefore  in  partial  opposition,  and  will  tend  to  reduce 
the  terminal  voltage.  On  the  other  hand,  if  the  armature  cur- 
rent leads,  the  e.m.f.  of  self-inductive  reactance  will  be  less  than 
90°  behind  the  generated  e.m.f.,  or  in  partial  conjunction  there- 
with, and  increase  the  terminal  voltage.  This  means  that  the 
e.m.f.  of  self-inductive  reactance  increases  the  terminal  voltage 
with  a  leading,  and  decreases  it  with  a  lagging  current,  or,  in 
other  words,  acts  in  the  same  manner  as  the  armature  reaction. 
For  this  reason  both  actions  can  be  combined  in  one,  and  repre- 
sented by  what  is  called  the  synchronous  reactance  of  the  alter- 
nator. In  the  following,  we  shall  represent  the  total  reaction 
of  the  armature  of  the  alternator  by  the  one  term,  synchronous 
reactance.  While  this  is  not  exact,  as  stated  above,  since  the 
reactance  should  be  resolved  into  the  magnetic  reaction  due  to 
the  magnetizing  action  of  the  armature  current,  and  the  electric 
reaction  due  to  the  self-induction  of  the  armature  current,  it  is 
in  general  sufficiently  near  for  practical  purposes,  and  well  suited 
to  explain  the  phenomena  taking  place  under  the  various  condi- 
tions of  load.  This  synchronous  reactance,  x,  is  occasionally  not 
constant,  but  is  pulsating,  owing  to  the  synchronously  varying 
reluctance  of  the  armature  magnetic  circuit,  and  the  field  mag- 
netic circuit;  it  may,  however,  be  considered  in  what  follows  as 
constant;  that  is,  the  e.m.fs.  generated  thereby  may  be  repre- 
sented by  their  equivalent  sine  waves.  A  specific  discussion  of 
the  distortions  of  the  wave  shape  due  to  the  pulsation  of  the  syn- 
chronous reactance  is  found  in  Chapter  XXVI.  The  synchron- 


ALTERNATING-CURRENT  GENERATOR          263 

ous  reactance,  x,  is  not  a  true  reactance  in  the  ordinary  sense  of 
the  word,  but  an  equivalent  or  effective  reactance.  Sometimes  the 
total  effects  taking  place  in  the  alternator  armature  are  repre- 
sented by  a  magnetic  reaction,  neglecting  the  self-inductive  re- 
actance altogether,  or  rather  replacing  it  by  an  increase  of  the 
armature  reaction  or  armature  m.m.f.  to  such  a  value  as  to 
include  the  self-inductive  reactance.  This  assumption  is  often 
made  in  the  preliminary  designs  of  alternators.  Further  dis- 
cussion of  the  relation  of  armature  reaction  and  self-induction 
see  "Theory  and  Calculation  of  Electrical  Circuits"  under 
"Reactance  and  Apparatus." 

187.  Let  EQ  =  generated  e.m.f.  of  the  alternator,  or  the  e.m.f. 
generated  in  the  armature-coils  by  their  rotation  through  the 
constant  magnetic  field  produced  by  the  current  in  the  field- 
spools,  or  the  open-circuit  voltage,  more  properly  called  the 
"  nominal  generated  e.m.f./'  since  in  reality  it  does  not  exist 
as  before  stated. 
Then 

EQ  =  V2  Trnf$  10~8; 
where 

n  =  total  number  of  turns  in  series  on  the  armature, 

/  =  frequency, 

$  =  total  magnetic  flux  per  field-pole. 
Let 

XQ  =  synchronous  reactance, 

r0  =  internal  resistance  of  the  alternator; 
then 

ZQ  .=  r0  H-  jxQ  =  internal  impedance. 

If  the  circuit  of  the  alternator  is  closed  by  the  external  im- 
pedance, 

Z  =  r  +  jx, 
the  current 


or, 

A/(T*O  +  02 
and,  the  terminal  voltage, 

E  =  /  Z  =  -c/o  —  /^o 


(TO+  r)  +j(xQ+  x) 


264         ALTERNATING-CURRENT  PHENOMENA 
or, 


E  = 


E, 


/I    _1_  9  r«r  +  X<&     ,    r°2 

V1  h2  r2  +  X2   +  - 


or,  expanded  in  a  series, 

7*o7*  -i-  X<& 
'   r2  +  z2 


, 

T 


4  (sr0  - 


!i 

si 


a 

/ 

s 

N\ 

-—  -», 

--^^^ 

/ 

' 

\ 
\ 

^ 

^ 

/ 

\ 

/ 

^< 

J^x 

\ 

/ 

X,, 

\ 

/ 

\ 

\ 

«, 

/ 

\ 

i 
\ 

\ 

i 

,y 

\ 

i 

\ 

i 

y 

\ 

t 

i 

FIELD  CHARACTERISTIC 
E0=2500,  Z<j=1-H0j 

d 

3 

/ 

3 

AMPS. 

FIG.  132. — Field  characteristic  of  alternator  on  non-inductive  load. 

As  shown,  the  terminal  voltage  varies  with  the  conditions  of 
the  external  circuit. 

188.  As  an  example  are  shown  in  Figs.  132-137,  at  constant 
generated  e.m.f., 

EQ  =  2500; 


ALTERNATING-CURRENT  GENERATOR 


265 


and  the  values  of  the  internal  impedance, 

ZQ  = 


with  the  current,  7,  as  abscissas,  the  terminal  voltages,  E, 
as  ordinates  in  full  line,  and  the  kilowatts  output,  =  72r,  in 
dotted  lines,  the  kilovolt-amperes  output,  =  IE,  in  dash-dotted 
lines,  for  the  following  conditions  of  external  circuit: 


*b 

°1 

\ 

22 

°0 

N, 

\ 

FIELD  CHARACTERISTIC 
E0=2500,ZrH-10j,-£=.75,°r60%P.F. 

\ 

18 
16 

•• 

<1Q 

8 
C 
4 
2 

n 

\ 

\ 

.• 

^ 

s 

s 

\ 

/ 

^S/! 

3< 

\ 

>/ 

"\ 

\ 

^ 

y 

^ 

'"' 

s 

\J 

/ 

^> 

y 

^ 

x 

\ 
\ 

1  / 

' 

\N 

\ 
^       \ 

% 

f 

\ 

w 

(/ 

s 

\ 

0   20   40   60   80   100  120  140  160  180  200  220  240  260  280 

AMPS. 

FIG.  133. — Field  characteristic  of  alternator  at  60  per  cent,  power-factor  on 

inductive  load. 

In  Fig.  132,  non-inductive  external  circuit,  x  =  0. 

In  Fig.  133,  inductive  external  circuit,  of  the  condition,  -   = 

+  0.75,  or  a  power-factor,  0.6. 

In  Fig.  134,  inductive  external  circuit,  of  the  condition,  r  =  0, 
or  a  power-factor,  0. 

In  Fig.  135,  external  circuit  with  leading  current,  of  the  condi- 
tion, x  =  —  0.75,  or  a  power-factor,  0.6. 

In  Fig.  136,  external  circuit  with  leading  current,  of  the  condi- 
tion, r  =  0,  or  a  power-factor,  0. 

In  Fig.  137,  all  the  volt-ampere  curves  are  shown  together  as 


266 


ALTERNATING-CURRENT  PHENOMENA 


complete  ellipses,  giving  also  the  negative  or  syn- 
chronous motor  part  of  the  curves. 
Such  a  curve  is  called  a  field  characteristic. 
As  shown,  the  e.m.f.  curve  at  non-inductive  load  is  nearly 
horizontal  at  open-circuit,  nearly  vertical  at  short-circuit,  and 
is  similar  to  an  arc  of  an  ellipse. 

With  reactive  load  the  curves  are  more  nearly  straight  lines. 
The  voltage  drops  on  inductive  load  and  rises  on  capacity  load. 


26 
24 

no 

\ 

\ 

\ 

FIELD  CHARACTERISTIC 
E0=2500,  Z5-1+1Oj.r=o,  9O°LAG 
1  2  T  =  0 

20 

18 

16 
(/> 

il" 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

X 

"\ 

"" 

X 

/ 

\ 

/. 

\ 

sy 

X 

V^x> 

\ 

8 
6 

4 
2 

° 

y 

/ 

X 

\ 

\ 

/ 

X 

\ 

N 

v 

/ 

\ 

\ 

\ 

/ 

x 

0 

/ 

X 

^ 

)       20      40       60       80      100     120     140     160     180     200     220     240     260  281 
AMPS. 

FIG.  134. — Field  characteristic  of  alternator  on  wattless  inductive  load. 


The  output  increases  from  zero  at  open-circuit  to  a  maximum, 
and  then  decreases  again  to  zero  at  short-circuit. 

189.  The  dependence  of  the  terminal  voltage,  E,  upon  the 
phase  relation  of  the  external  circuit  is  shown  in  Fig.  138,  which 
gives,  at  impressed  e.m.f.,  EQ  =  2500  volts,  and  the  currents, 
/  =  50,  100,  150,  200,  250  amp.,  the  terminal  voltages,  E,  as 
ordinates,  with  the  inductance  factor  of  the  external  circuit 

=r,  as  abscissas. 


ALTERNATING-CURRENT  GENERATOR 


267 


190.  If  the  internal  impedance  is  negligible  compared  with 
the  external  impedance,  then,  approximately, 


E  = 


VOo  +  r)2  +  (x0  +  z)2 


that  is,   an  alternator  with  small  internal  resistance  and    syn- 
chronous reactance  tends  to  regulate  for  constant-terminal  voltage. 


VOLTS 


^ ^870        368Q-^>> 


3600 

^ 

< 

s 

^ 

^ 

* 

\ 

y,     3200 

H 

0    2800 

_i 

2 

1200  2400 
1000  2000 
800  1600 
600  1200 
400   800 

200   400 
100 
0      0 

s* 

^ 

/ 

^r 

X 

/ 

FIELD  CHARACTERISTIC 

/ 

E0=2500,  Z0=1+10j,7=r75or  60%P.F. 

.' 

^»^- 

~~^ 

X 

,'' 

/ 

/' 

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! 

/ 

/ 

s 

j. 

..  ( 

^ 

/ 

^ 

/ 

n 
/ 

\ 

s 

X 

* 

^ 

/ 

f 

i 
/ 

• 

/ 

& 

s* 

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7 

/ 

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J 

s 

/ 

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t 

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/ 

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f 

x^ 

X 

^ 

,./ 

-*" 

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K-- 

)           40           80         120          160          200         240          280         320          360 

AMPERES 

FIG.  135. — Field  characteristic  of  alternator  at  60  per  cent,  power-factor  on 

condenser  load. 

Every  alternator  does  this  near  open-circuit,  especially  on 
non-inductive  load. 

Even  if  the  synchronous  reactance,  XQ,  is  not  quite  negli- 
gible, this  regulation  takes  place,  to  a  certain  extent,  on  non- 
inductive  circuit,  since  for  x  =  0, 


E  = 


2  r 


and  thus  the  expression  of  the  terminal  voltage,  E,  contains 


268         ALTERNATING-CURRENT  PHENOMENA 


the  synchronous  reactance,  X0,  only  as  a  term  of  second  order  in 
the  denominator. 

On  inductive  circuit,  however,  x0  appears  in  the  denominator 


44 
42 
40 
38 
36 
34 
32 
30 
28 
26 
24 
{/>      22 

l^o 

§8  is 

x  * 
16 

14 

12 
10 

8 
6 
4 
2 

'1 

i 

FIELD  CHARACTERISTIC 
E<j=25OO,    Z0=110j, 
r=o,  90°Leading  Current 

I2r=a 

! 

1 
i 

t 

1 

/ 

/ 

i 

/ 

i 

/ 

I 

/ 

i 

/ 

1 

/ 

1 

/ 

7 

& 

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1 

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s 

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f 

i 

-f 

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/ 

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VI 

/ 

/ 

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/ 

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/ 

/ 

''/ 

/ 

/ 

// 

'/ 

^ 

4 

S 

12       14       16       18      20       22      24       26     28 
xlOO=AMPS. 


024         6        8       10 

FIG.  136. — Field  characteristic  of  alternator  on  wattless  condenser  load. 


as  a  term  of  first  order,  and  therefore  constant-potential  regu- 
lation does  not  take  place  as  well. 

With   a   non-inductive  external   circuit,   if  the   synchronous 


ALTERNATING-CURRENT  GENERATOR 


269 


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350 

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34 
32 
3C 
2£ 
2( 
& 
25 
2( 

tie 

8n 

XJJ 

1C 

f 
1 

4 

s 

FIG.  137.  —  Field  characteristic  of  alternator. 

i 

/ 

/ 

/ 

^ 

Eo- 

,25 
^50 
=10( 

15C 
on 

00,Z0=H10j 
Amps. 
)     " 
)     " 
0    " 
0   " 

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// 

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1  = 
1  = 
1= 

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l 

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1    .9    .8     .7    .6    .5    .4    .3    .2    .1     0  -1  -2  -.3  -.4  -.5  -.6  -.7  -.8  -.9  -1 

X 

r2-f  a;2 


FIG.  138.  —  Regulation  of  alternator  on  various  loads. 


270         ALTERNATING-CURRENT  PHENOMENA 

reactance,  x0,  of  the  alternator  is  very  large  compared  with  the 

external  resistance,  r, 

current 


approximately,  or  constant;  or,  if  the  external  circuit  contains 
the  reactance,  x, 

T  -        -^o  J^o 


approximately,  or  constant. 

In  this  case,  the  terminal  voltage  of  a  non-inductive  circuit  is 


approximately  proportional  to  the  external  resistance.     In  an 
inductive  circuit, 


approximately  proportional  to  the  external  impedance. 

191.  That  is,  on  a  non-inductive  external  circuit,  an  alter- 
nator with  very  low  synchronous  reactance  regulates  for  con- 
stant-terminal voltage,  as  a  constant-potential  machine,  an 
alternator  with  a  very  high  synchronous  reactance  regulates  for 
a  terminal  voltage  proportional  to  the  external  resistance  as  a 
constant-current  machine. 

Thus,  every  alternator  acts  as  a  constant-potential  machine 
near  open-circuit,  and  as  a  constant-current  machine  near  short- 
circuit.  Between  these  conditions,  there  is  a  range  where  the 
alternator  regulates  approximately  as  a  constant-power  machine, 
that  is,  current  and  e.m.f.  vary  in  inverse  proportion,  as  between 
130  and  200  amp.  in  Fig.  *132. 

The  modern  alternators  are  generally  more  or  less  machines 
of  the  first  class;  the  old  alternators,  as  built  by  Jablockkoff, 
Gramme,  etc.,  were  machines  of  the  second  class,  used  for  arc 
lighting,  where  constant-current  regulation  is  an  advantage. 

Very  high-power  steam-turbine  alternators  are  now  again  built 
with  fairly  high  reactance,  for  reasons  of  safety. 

Obviously,  large  external  reactances  cause  the  same  regula- 


ALTERNA  TING-C  URRENT  GENERA  TOR         271 

tion  for  constant  current  independently  of  the  resistance,   r, 
as  a  large  internal  reactance,  XQ. 
On  non-inductive  circuit,  if 

r-     ,         E° 

V(r  +  r0)2  +  z02 

and 


the  output  is 

p  =  IE  =  ^°2~ 

and 

dP  _      X02  -  r2  +  r02 

Hence,  if 
or 

'  i-°- 

the  power  is  a  maximum,  and 

BJ 

P  = 


# 

and 


2  {^o  +  r0} 

Eo 


I  = 


Therefore,  with  an  external  resistance  equal  to  the  internal 
impedance,  or,  r  =  z0  =  Vr02  +  #02, tne  output  of  an  alternator 
is  a  maximum,  and  near  this  point  it  regulates  for  constant 
output;  that  is,  an  increase  of  current  causes  a  proportional 
decrease  of  terminal  voltage,  and  inversely. 

The  field  characteristic  of  the  alternator  shows  this  effect 
plainly. 


CHAPTER  XXII 
ARMATURE  REACTIONS  OF  ALTERNATORS 

192.  The  change  of  the  terminal  voltage  of  an  alternating 
current  generator,  resulting  from  a  change  of  load  at  constant 
field  excitation,  is  due  to  the  combined  effect  of  armature 
reaction  and  armature  self-induction.  The  counter  m.m.f.  of 
the  armature  current,  or  armature  reaction,  combines  with  the 
impressed  m.m.f.  or  field  excitation  to  the  resultant  m.m.f., 
which  produces  the  resultant  magnetic  field  in  the  field  poles 
and  generates  in  the  armature  an  e.m.f.  called  the  ''virtual 
generated  e.m.f./'  since  it  has  no  actual  existence,  but  is  merely 
a  mathematical  fiction.  The  counter  e.m.f.  of  self-induction  of 
the  armature  current,  that  is,  e.m.f.  generated  by  the  armature 
current  by  a  local  magnetic  flux,  combines  with  the  virtual 
generated  e.m.f.  to  the  actual  generated  e.m.f.  of  the  armature, 
which  corresponds  to  the  magnetic  flux  in  the  armature  core. 
This  combined  with  the  e.m.f.  consumed  by  the  armature  resist- 
ance gives  the  terminal  voltage. 

In  most  cases  the  effect  of  armature  reaction  and  of  self- 
induction  are  the  same  in  character,  and  so  both  effects  usually 
are  contracted  in  one  constant;  for  purposes  of  design,  frequently 
the  self-induction  is  represented  by  an  increase  of  the  armature 
reaction,  that  is,  an  effective  armature  reaction  used  which  com- 
bines the  effect  of  the  true  armature  reaction  and  the  armature 
self-induction.  That  is,  instead  of  the  counter  e.m.f.  of  self- 
induction,  a  counter  m.m.f.  is  used,  which  would  produce  the 
magnetic  flux  which  would  generate  the  e.m.f.  of  self-induction. 
For  theoretical  investigations  usually  the  armature  reaction  is 
represented  by  an  effective  self-induction,  that  is,  instead  of  the 
counter  m.m.f.  of  the  armature  reaction,  the  e.m.f.  considered, 
which  would  be  generated  by  the  magnetic  flux,  which  the  arma- 
ture reaction  would  produce.  That  is,  both  effects  are  com- 
bined in  an  effective  reactance,  the  "synchronous  reactance." 

While  armature  reaction  and  self-inductance  are  similar  in 

272 


ARMATURE  REACTIONS  OF  ALTERNATORS    273 

effect,  in  some  cases  they  differ  in  their  action;  the  e.m.f.  of 
self-inductance  is  instantaneous,  that  is,  appears  and  disappears 
with  the  current  to  which  it  is  due.  The  effect  of  the  armature 
reaction,  however,  requires  time;  the  change  of  the  magnetic  field 
resulting  from  the  combination  of  the  counter  m.m.f.  of  arma- 
ture reaction  with  the  impressed  m.m.f.  of  field  excitation  occurs 
gradually,  since  the  magnetic  field  flux  interlinks  with  the  field 
winding,  and  any  sudden  change  of  the  field  generates  an  e.m.f. 
in  the  field  circuit,  which  temporarily  increases  or  decreases  the 
field  current,  and  so  retards  the  change  of  the  field  flux.  So,  for 
instance,  a  sudden  increase  of  load  results  in  a  simultaneous 
increase  of  the  counter  e.m.f.  of  self-induction  and  counter 
m.m.f.  of  armature  -reaction.  With  the  armature  reaction 
demagnetizing  the  field,  the  field  flux  begins  to  decrease,  and 
thus  generates  an  e.m.f.  in  the  field-exciting  circuit,  which 
increases  the  field  current  and  retards  the  decrease  of  field 
flux,  so  that  the  field  flux  adjusts  itself  only  gradually  to  the 
change  of  circuit  conditions,  at  a  rate  of  speed  depending  upon 
the  constants  of  the  field-exciting  circuit,  etc. 

The  extreme  case  hereof  takes  place  when  suddenly  short- 
circuiting  an  alternator;  at  the  first  moment  the  short-circuit 
current'  is  limited  only  by  the  self-inductance,  and  the  magnetic 
field  still  has  full  strength,  the  field-exciting  current  has  greatly 
increased  by  the  e.m.f.  generated  in  the  field  circuit  by  the  arma- 
ture reaction.  Gradually  the  field-exciting  current  and  there- 
with the  field  magnetism  die  down  to  the  values  corresponding 
to  the  short-circuit  condition.  Thus  the  momentary  short- 
circuit  current  of  an  alternator  is  far  greater  than  the  perma- 
nent short-circuit  current;  many  times  in  a  machine  of  low 
self-induction  and  high  armature  reaction,  as  a  low-frequency, 
high-speed  alternator  of  large  capacity;  relatively  little  in  a 
machine  of  low  armature  reaction  and  high  self-induction,  as  a 
high-frequency  unitooth  alternator. 

193.  Graphically,  the  internal  reactions  of  the  alternating- 
current  generator  can  be  represented  as  follows: 

Let  the  impressed  m.m.f.,  or  field  excitation,  F0,  be  repre- 
sented by  the  vector  OFo,  in  Fig.  139,  chosen  for  convenience 
as  vertical  axis.  Let  the  armature  current,  /,  be  represented  by 
vector  01.  This  current,  /,  gives  armature  reaction  FI  =  nl, 
where  n  =  number  of  effective  turns  of  the  armature,  and  is  repre- 
sented by  the  vector,  OF1}  with  the  two  quadrature  components, 

18 


274         ALTERNATING-CURRENT  PHENOMENA 

OF'i,  in  line  with  the  field  m.m.f.,  OF0 — and  usually  opposite 
thereto — and  OF,",  in  quadrature  with  OF0. 

OFo  combined  with  OFi  gives  the  resultant  m.m.f.,  OF,  with 
the  quadrature_components,  OF'  =  OFQ  —  OF'i,  and  OF". 

The  m.m.f.,  OF,  produces  a  magnetic  flux,  0«J>,  and  this  gener- 
ates an  e.m.f.,  OE2,  in  the  armature  circuit,  90°  behind  OF  in 
phase,  the  virtual  generated  e.m.f. 


FIG.  139. 


The  armature  self-induction  consumes  an  e.m.f.,  Q#3,  90° 
ahead  of  the  current,  thus,  subtracted  vectorially  from  OE2, 
gives  the  actual  generated  e.m.f.,  OEi.  

The  armature  resistance,  r,  consumes  an  e.m.f.,  OE*,  in  phase 
with  the  current,  which  subtracts  vectorially  from  the  actual 
generated  e.m.f.,  and  thus  gives  the  terminal  voltage,  OE. 

194.  Analytically,  these  reactions  are  best  calculated  by  the 
symbolic  method. 


ARMATURE  REACTIONS  OF  ALTERNATORS    275 

Let  the  impressed  m.m.f.,  or  field-excitation,  FQ,  be  chosen  as 
the  imaginary  axis,  hence  represented  by 

^o  =  +  J/o  (1) 

Let 

I  =  i\  —  jiz  =  armature  current.  (2) 

The  m.m.f.  of  the  armature  then  is 


Fl  =  nl  =  n(ii  -jit)  (3) 

where 

n  =  number  of  effective  armature  turns, 

and  the  resultant  m.m.f.  then  is 

F  =  Fo  +  F  i  =  j(/o  -  nit)  +  nil.  (4) 

If,  then, 

(P  =  magnetic   permeance   of  the  structure,  that   is,  magnetic 
flux  divided  by  the  ampere-turns  m.m.f.  producing  it, 

$ 
(P  =  j,,  or,  $  =  (PF  =  j(P(/0  -  nit)  +  (Pm'i.  (5) 

The  e.m.f.  generated  by  the  magnetic  flux  &  in  the  armature 

is  e2  =  27r/n<f>10-8,  (6) 

where/  =  frequency. 

Denoting  2  irfn  10  ~  8  by  a  we  have,  (7) 

€2  =  a  3>  (8) 

and  since  the  generated  e.m.f.  is  90°  behind  the  generating  flux,  in 
symbolic  expression, 


(9) 
hence,  substituting  (5)  in  (9), 

—  niz)  —  ja(?nii,  (10) 


the  virtual  generated  e.m.f. 

The  e.m.f.  consumed  by  the  self-inductive  reactance  of  the 
armature  circuit  is, 

E3  =  jxl  =  jxii  +  xiz;  (11) 


and  therefore,  the  actual  generated  e.m.f. 

Ei  =  E2  -  E3 

=  {a(P/o  -  (a(Pw  +  x)i2}  -  jii(a(?n  +  x).       (12) 


276         ALTERNATING-CURRENT  PHENOMENA 

The  e.m.f.  consumed  by  the  armature  resistance,  r,  is 

#4  =  rl  =  rii  -  jriz; 
hence,  the  terminal  voltage, 

E  =  El-  E, 

=  {a&fo  —  (a(?n  +  x)iz  —  rii}  —  j{ii(a(S>n  +  x)  —  riz}.  (14) 

195.  It  is 

/o  =  field  m.m.f.;  hence 

3>o  =  (P/o  =  magnetic  flux,  which  would  be  produced  by  the 
field  excitation,  /0,  if  the  magnetic  permeance  at  this  m.m.f.,  /0, 
were  the  same,  (P,  as  at  the  m.m.f.,  F  —  that  is,  if  the  magnetic 
characteristic  would  not  bend  between  /o  and  F,  due  to  mag- 
netic saturation,  or  in  other  words,  when  neglecting  saturation, 
and  therefore  e0  =  a(P/0(15)  =  e.m.f.  generated  in  the  armature 
by  the  field  excitation,  when  neglecting  magnetic  saturation,  or 
assuming  a  straight  line  saturation  curve. 

eo  is  called  the  "nominal  generated  e.m.f.  of  the  machine." 

ni  =  armature  m.m.f.  ;  therefore, 
(?ni  =  magnetic  flux  produced  thereby,  and, 
a(?ni  =  e.m.f.  generated  in  the  armature  by  the  magnetic  flux  of 
armature  reaction,  hence, 

a(9n  =  Xi 

=  effective  reactance,  representing  the  armature  reaction, 

and  XQ  =  a(?n  +  x  f  (16) 

=  synchronous  reactance,  that  is,  the  effective  reactance 

representing  the  combined  effect  of  armature  self-induction  and 

armature  reaction. 

Substituting  (15)  and  (16)  in  (14)  gives, 

E  =  (g0  -  x0iz  -  rii)  -  j(xQii  -  riz)  (17) 


It  follows  herefrom: 

In  an  alternating-current  generator,  the  combined  effect  of 
armature  reaction  and  self-induction  can  be  represented  by  an 
effective  reactance,  the  synchronous  reactance,  XQ,  which  consists 
of  the  two  components: 

x0  =  x  H-  xi  (18) 

where, 

x  =  true  self-inductive  reactance  of  the  armature  circuit. 
x\  =  a&n  =  effective  reactance  of  armature  reaction,          (19) 


ARMATURE  REACTIONS  OF  ALTERNATORS     277 

and  the  nominal  generated  e.m.f., 

eQ  =  a(P/0;  (15) 

where, 

n   —  number  of  armature  turns,  effective, 

fo  =  field  excitation,  in  ampere-turns, 

a    =  2  irfwlO-8.  (7) 

(P  =  magnetic  permeance  of  the  field  structure  at  a  magnetic 
flux  in  the  field-poles  corresponding  to  the  virtual  generated 
e.m.f.,  E2. 

The  introduction  of  the  term  " synchronous  reactance,"  x0, 
and  "nominal  generated  e.m.f.,"  e0,  is  hereby  justified,  when 
dealing  with  the  permanent  condition  of  the  electric  circuit. 
The  case  of  the  transient  phenomena  of  momentary  short- 
circuit  currents,  etc.,  is  discussed  in  a  chapter  on  "Transient 
Phenomena  and  Oscillations/'  section  I. 

It  must  be  understood  that  the  "nominal  generated  e.m.f.," 
e0,  in  an  actual  machine,  in  which  the  magnetic  characteristic 
bends  due  to  the  approach  to  magnetic  saturation,  is  not  the 
voltage  generated  by  the  field  excitation  /0  at  open-circuit,  but 

is  the  voltage  which  would  be  generated,  if  at  excitation,  /0,  the 

$ 
magnetic  permeance,  (P  =  ^  were  the  same  as  at  the  actual  flux 

existing  in  the  machine — that  is,  if  the  magnetic  characteristic 
would  continue  in  a  straight  line  passing  through  the  origin  when 
prolonged. 

The  equation  (17)  may  also  be  written 

E   =  eQ  -  Z07;  (20) 

where, 

ZQ  =  r  +  jx0  =  synchronous  impedance  of  the  alternator. 

/    =  ii   —  jiz, 
or,  more  generally 

E  =  E0  -  Z0I,  (22) 

and  so  is  the  equation  of  a  circuit,  supplied  by  the  e.m.f.,  E0, 
with  the  current,  /,  over  the  impedance,  ZQ,  as  has  been  discussed 
in  the  chapter  on  resistance,  inductive  reactance  and  conden- 
sive  reactance. 


278         ALTERNATING-CURRENT  PHENOMENA 

An  alternator  so  is  equivalent  to  an  e.m.f.,  E0,  the  nominal 
generated  e.m.f.,  supplying  current  over  an  impedance,  Z0,  the 
synchronous  impedance. 

196.  In  theoretical  investigations  of  alternators,  the  syn- 
chronous reactance,  x0,  is  usually  assumed  as  constant,  and  has 
been  assumed  so  in  the  preceding. 

In  reality,  however,  this  is  not  exactly,  and  frequently  not 
even  approximately  correct,  but  the  synchronous  reactance  is 
different  in  different  positions  of  the  armature  with  regard  to  the 
field.  Since  the  relative  position  of  the  armature  to  the  field 
varies  with  the  armature  current,  and  with  the  phase  angle  of 
the  armature  current,  the  regulation  curve  of  the  alternator,  and 
other  characteristic  curves,  when  calculated  under  the  assump- 
tion of  constant  synchronous  reactance,  may  differ  considerably 
from  the  observed  curves,  in  machines  in  which  the  synchronous 
reactance  varies  with  the  position  of  the  armature. 

The  two  components  of  the  synchronous  reactance  are  the  self- 
inductive  reactance,  and  the  effective  reactance  of  armature 
reaction.  The  self-inductive  reactance  represents  the  e.m.f. 
generated  in  the  armature  by  the  local  field  produced  in  the 
armature  by  the  armature  current.  The  magnetic  reluctance 
of  the  self-inductive  field  of  the  armature  coil,  however,  is,  in 
general,  different  when  this  coil  stands  in  front  of  a  field-pole, 
and  when  it  stands  midway  between  two  field-poles,  and  the 
self-inductive  reactance  so  periodically  varies,  between  two 
extreme  values,  representing  respectively  the  positions  of  the 
armature  coils  in  front  of,  and  midway  between  the  field-poles, 
that  is,  pulsates  with  double  frequency,  between  a  value,  x', 
corresponding  to  the  position  in  front,  and  a  value,  x",  corre- 
sponding to  a  position  midway  between  the  field-poles.  Depend- 
ing upon  the  structure  of  the  machine,  as  the  angle  of  the  pole 
arc,  that  is,  the  angle  covered  by  the  pole  face,  either  x'  or  x" 
may  be  the  larger  one. 

The  effective  reactance  of  armature  reaction,  xi,  corresponds 
to  the  magnetic  flux,  which  the  armature  would  produce  in  the 
field-circuit.  With  the  armature  coil  facing  the  field-pole,  that 
is,  in  a  nearly  closed  magnetic  field-current,  x\,  therefore  is 
usually  far  greater  than  with  the  armature  coil  facing  midway 
between  the  field-poles,  in  a  more  or  less  open  magnetic  circuit. 
Hence,  Xi,  also  varies  between  two  extreme  values,  x\  and  Xi", 
corresponding  respectively  to  the  position  in  line  with,  and  in 


ARMATURE  REACTIONS  OF  ALTERNATORS    279 

quadrature  with,  the  field-poles.  In  this  case,  usually  xi  is 
larger  than  #/'. 

Since  x\  =  a(?n,  where  (P  =  magnetic  permeance,  (P  varies 
between  (P',  corresponding  to  the  position  of  the  armature  coil 
opposite  the  field-poles,  and  (P",  corresponding  to  the  position 
of  the  armature  coil  midway  between  the  field-poles.  Usually 
(9'  is  far  larger. 

This  means  that  the  two  components  of  the  resultant  m.m.f. 
F:  Fi,  in  line  with,  and  F"  in  quadrature  with,  the  field-poles, 
act  upon  magnetic  circuits  of  very  different  permeance,  (P'  and 
(P",  and  the  components  of  magnetic  flux,  due  to  F'  and  F" 
respectively,  are 


$>"  =  <s>"F". 

The  two  components  of  magnetic  flux,  <£>'  and  <£",  therefore 
are  in  general,  not  proportional  to  their  respective  m.m.fs.  Ff  and 
F",  and  the  resultant  flux,  <£,  accordingly  is  not  in  line  with  the 
resultant  m.m.f.,  F,  but  differs  therefrom  in  direction,  being 
usually  nearer  to  the  center  line  of  the  field-poles.  That  is, 
the  resultant  magnetic  flux,  <£,  is  more  nearly  in  line  with  the 
impressed  m.m.f.  of  field  excitation,  F0,  than  the  resultant 
m.m.f.,  F,  is  —  or  in  other  words  —  the  magnetic  flux  is  shifted 
by  the  armature  reaction  less  than  the  resultant  m.m.f.  is  shifted. 

197.  To  consider,  in  the  investigation  of  the  armature  reactions 
of  an  alternator,  the  difference  of  the  magnetic  reluctance  of  the 
structure  in  the  different  directions  with  regard  to  the  field,  that 
is,  the  effect  of  the  polar  construction  of  the  field,  or  the  use  of 
definite  polar  projections  in  the  magnetic  field,  the  reactions 
of  the  machine  must  be  resolved  into  two  components,  one  in 
line  and  the  other  in  quadrature  with  the  center  line  of  the  field- 
poles,  or  the  direction  of  the  impressed  m.m.f.  or  field-excitation, 
F* 

Denoting  then  the  components  in  line  with  the  field-poles 
or  parallel  with  the  field-excitation,  FQ,  by  prime,  as  /',  F',  etc., 
and  the  components  facing  midway  between  the  field-poles,  or 
in  quadrature  position  with  the  field-excitation,  FQ,  by  second, 
as  /",  F",  the  diagram  of  the  alternator  reactions  is  modified 
from  that  given  in  Fig.  139. 

Choosing  again,  in  Fig.  140,  the  impressed  m.m.f.  or  field- 
excitation,  FQ,  as  vertical  vector  OFQ,  the  current,  OI,  consists 


280 


ALTERNATING-CURRENT  PHENOMENA 


of  the  component,  01' ',  in  line  with  F0,  or  vertical,  and  OI"  in 
quadrature  with  F0,  or  horizontal.  The  armature  reaction, 
OFi,  gives  the  components,  OFi  and  OFi",  and  the  resultant 
m.m.f.  therefore  consists  of  two  components,  OF'  =  OFQ  — 
OF/,  and  OF"  =  OFi". 


FIG.  140. 

Let  now 
<?'    =  permeance  of  the  field  magnetic  circuit;  (23) 

(P"  =  permeance  of  the  magnetic  circuit  through  the  armature 
in  quadrature  position  to  the  field-poles;  (24) 

the  components  of  the  resultant  magnetic  flux  are, 

$'  =  ®'p't  represented  by  0<i>';  and   $"  =  <S>"Fn ',  represented 
by  0*", 

and  the  Resultant  magnetic  flux,  by_cpmbination  of  O&  and 
O$",  is  0$,  and  is  not  in  line  with  OF,  but  differs  therefrom, 
being  usually  nearer  to  OFo. 


ARMATURE  REACTIONS  OF  ALTERNATORS    281 

The  virtual  generated  e.m.f.  is 

E2  =  a$, 

and  represented  by  OE%,  90°  behind  0<I>. 
Let  now 

xf  =  self-inductive  reactance  of  the  armature  when  facing 
the  field-poles,  and  thus  corresponding  to  the  compo- 
nent, 7',  of  the  current,  (25) 
and 

x"  =  self-inductive  reactance  of  the  armature  when  facing 
midway  between  the  field-poles,  and  thus  corresponding 
to  the  component,  7",  of  the  current.  (26) 

Then 

E'z  =  xT  =  e.m.f.   consumed  by  the  self-induction  of  the 

current  component,  I', 
and 

E"z  =  x"I"  —  e.m.f.  consumed  by  the  self-induction  of  the 
current  component,  I". 

E'3  is  represented  byjvector  OE'z,  90°  ahead  of  07',  and  E"3  is 
represented  by  vector  OE"z,  90°  ahead  of  01".  The  resultant 
e.m.f.  of  self-induction  then  is  given  by  the  combination  of  OE'z 
and  OE"z,  as  OE*.  It  is  not  90°  ahead  of  07,  but  either  more 
or  less.  In  the  former  case,  the  self-induction  consumes  power, 
in  the  latter  case,  it  produces  power.  That  is,  in  such  an  arma- 
ture revolving  in  the  structure  of  non-uniform  reluctance,  the 
e.m.f.  of  self-induction  is  not  wattless,  but  may  represent  con- 
sumption, or  production  of  power,  as  "reaction  machine."  (See 
"Calculation  of  Electrical  Apparatus.") 

__Subtracting  vectorially  OEs  from  the  virtual  generated  e.m.f. 
OE%,  gives  the  actual  generated  e.m.f.,  OEi,  and  subtracting 
therefrom  the  e.m.f.  consumed  by  the  armature  resistance,  OE*, 
in  phase  with  the  current,  07,  gives  the  terminal  voltage,  OE. 

198.  Here  the  diagram  has  been  constructed  graphically,  by 
starting  with  the  field-excitation,  Fo,  the  armature  current,  7, 
and  the  phase  angle  between  the  armature  current,  7,  and  the 
field-excitation,  F0 — that  is,  the  angle  between  the  position  in 
which  the  armature  current  reaches  its  maximum,  and  the  direc- 
tion of  the  field-poles.  This_angle,  however,  is_unknown.  Usu- 
ally the  terminal  voltage,  OE,  the  current,  07,  and  the  angle, 


282 


ALTERNATING-CURRENT  PHENOMENA 


EOI,  between  current  and  terminal  voltage  are  given.  From 
these  latter  quantities,  however,  the  diagram  cannot  be  con- 
structed, since  the  position  of  the  field-excitation,  FO,  and  so  the 
directions,  in  which  the  electric  quantities  have  to  be  resolved 
into  components,  are  still  unknown,  when  starting  the  construc- 
tion of  the  diagram. 

That  is,  as  usually,  the  graphical  representation  affords  an 

insight  into  the  inner  relations 
of  the  phenomena,  but  not  a 
method  for  their  numerical  cal- 
culations, and  for  the  latter 
purpose,  the  symbolic  method 
is  required. 
Let 

EQ  =  nominal  generated 
e.m.f.,   or   e.m.f.   corresponding 

to  the  field-excitation,  FQ,  on  a 

straight  line  continuation  of  the 
magnetic  characteristic  from  the 
•  *  actual  value  of  the  field  onward 

— as  shown  by  Fig.  141. 
The  impressed  m.m.f.,  or  field  excitation,  is  then  given  by 


JF0. 


(27) 


Let 


/  =  I'  +  I"  =  armature  current,  (28) 

where  the  component,  7',  is  in  line,  the  component,  I",  in  quad- 
rature with:  jFQ. 

If  n  =  number  of  effective  armature  turns,  the  m.m.f.  of  the 
armature  current,  /,  or  the  armature  reaction,  then  is 

F,  =  nl,  (29) 

with  its  components,  in  phase  and  in  quadrature  with  the  field; 


Fi"  =  w7"; 
and  the  components  of  the  resultant  m.m.f.  then  are 

F"    =  rc7"; 


(30) 


(31) 


ARMATURE  REACTIONS  OF  ALTERNATORS    283 

and  the  resultant 

F  =  JF<>  +  nl'  +  nl".  (32) 

The  components  of  the  magnetic  flux,  in  line  and  in  quadrature 
with  jF0,  then  are 


=  (P''Fo  +  n/O;  (33) 

$"  =  <p"F" 

=  <P"'n7";  (34) 

hence,  the  resultant  magnetic  flux 

$  =  $'  -f  $" 

=  (P'CjFo  +  nl')  +  (P^nJ"  (35) 

The  e.m.f.  generated  by  this  magnetic  flux,  $,  or  the  virtual 
generated  e.m.f.  is 


=  -  a(?'(FQ  +  jnl1)  +  -  ja&"nl".  (36) 

The  e.m.f.  consumed  by  the  self-inductive  reactance,  x',  of 
the  current  component,  /',  is, 


E'3  =  jx'I',  (37) 

the  e.m.f.  consumed  by  the  self-inductive  reactance,  x",  of  the 
current  component,  /",  is 

E"9  =  jx"I",  (38) 

and  the  total  e.m.f.  consumed  by  self-induction  thus  is 

#3  =  j(x'I'  +  x"I");  (39) 

hence,  the  actual  generated  e.m.f. 
EI  =  EZ  —  EZ 

=  a(?'F0  -  jl'(a(?'n  +  x')  -  jl"(a(?"n  +  x").  (40) 

The  e.m.f.  consumed  by  the  resistance,  r,  is 

E4  =  rl 

=  rl'  +  rl";  (41) 


284         ALTERNATING-CURRENT  PHENOMENA 

hence,  the  terminal  voltage  of  the  machine  is 

E  ==  E\  —  E\ 

=  a(P'F0  -I'(r+  j(a(S>'n  +  *') }  -  I"  (r  +  j  (a(P"n  +  x") }.  (42) 
In  this  equation  of  the  terminal  voltage, 

x'0  =  a(P'w  -f  x', 


x"Q  =  a(P"n  +  x",  \  (43) 

are  effective  reactances,  corresponding  to  the  two  quadrature 
positions;  that  is 

X'Q  =  synchronous  reactance  corresponding  to  the  position 
of  the  armature  circuit  parallel  to  the  field  circuit ;  (44a) 

x"o  =  synchronous  reactance  corresponding  to  the  position  of 
the  armature  circuit  in  quadrature  with  the  field  circuit;  (446) 

a(S>'F0  is  the  e.m.f.  which  would  be  generated  by  the  field 
excitation,  F0,  with  the  permeance,  <?',  in  the  direction  in  which 
the  field  excitation,  F0,  acts,  that  is 

EQ  =  aCP'Fo  =  nominal  generated  e.m.f.          (45) 
and  it  is:  terminal  voltage, 

E  =  Eo  -  l'(r  +  jz'o)  -  l"(r  +  jx"0).        (46) 

That  is,  even  with  an  heteroform  structure,  as  a  machine 
with  definite  polar  projections,  the  armature  reaction  and 
armature  self-induction  can  be  combined  by  the  introduction 
of  the  terms  "nominal  generated  e.m.f."  and  "synchronous 
reactance,"  as  defined  above,  except  that  in  this  case  the  syn- 
chronous reactance,  XQ,  has  two  different  values,  X'Q  and  x"o, 
corresponding  respectively  to  the  two  main  axes  of  the  magnetic 
structure,  in  line  and  in  quadrature  with  the  field-poles. 

199.  In  the  equation  (46),  E,  E0,  I'  and  /"  are  complex 
quantities,  and 

I"  is  in  phase  with  EQ, 

I'  is  in  quadrature  behind  E0,  and  so  behind  I": 
hence,  /'  can  be  represented  by 

/'  =  -  jtl",  (47) 


ARMATURE  REACTIONS  OF  ALTERNATORS     285 
where  t  =  ratio  of  numerical  values  of  /"  and  /',  that  is 

t  =  —  =  tan  B  (48) 

and 

8=  angle   of   lag   of   current,  /,   behind  nominal  generated 
e.m.f.,  EQ.     Then 

i  =  r  +  /"  =  /"(i  -jt)9 


or 


r  and 


-jt  -  -  r     i+# 

Substituting  these  values  (50)  in  equation  (46)  gives 


(49) 
(50) 


In  this  equation,  EQ  leads  /  by  angle  6. 
Hence,  choosing  the  current,  /,  as  zero  vector, 

7  =  i,  (52) 

the  e.m.f.,  E0,  which  leads  i  by  angle,  6,  can  be  represented  by 


EQ  =  60  (cos  0  -f-  j  sin  0), 
or,  since  by  equation  (48), 


(53) 


Oil!      I/       -                           ,  CfcUU      UUS      17       -                       y  

V04J 

€°             -fl  eoV^+^2. 

(55) 

.   °           ^/l   _j_  ^2    ^        "•     ^"          1    —  ji 

Substituting  (52)  and  (55)  in  equation  (51),  gives 

eoV'l  +  ^2  —  *  {  (T*  +  ja5"o)  —  j<  (r  +  jx;0)  } 

(56) 

1  —  jt 

Let 

TTT                         1 

JG/    =    61   -p  Jc/2, 

(57) 

where 

-2    =tan0', 

(58) 

and 


0'  «=  angle  of  lag  of  current,  i,  behind  terminal  voltage,  E,    (59) 


ALTERNATING-CURRENT  PHENOMENA 

substituting  (57)  in  (56)  and  transposing, 

eo  Vl+t*  ~  (ei+jes)  (1  -jt)  -i  { (r+jx"0)  -  jt  (r+jx'o) }  =  0,   (60) 
or,  expanded, 
{e0\/lJrf-ei-te2-i(r+tx'0)}+j{tei-ei+i(tr-x"0)}  =0.   (61) 

As  the  left  side  is  a  complex  quantity,  and  equals  zero,  the  real 
part  as  well  as  the  imaginary  part  must  be  zero,  and  equation  (61) 
so  resolves  into  the  two  equations 

e<>  Vl  +  t2  -  e,  -  te2  -i(r  +  te'0)  =  0,  (62) 

tei  -  e2  +  i  (tr  -  x"0)  =  0.  (63) 

From  equation  (63)  follows 

=  e,  -Ks'V'.  (     ^ 

ei  +  n 

Substituting  (64)  in  (62),  and  expanding,  gives 

*  =  (€'  +  ?*+  nvtff+y*'  (65) 


(66) 


That  is,  if 

X'Q   —  synchronous  reactance  in  the  direction  of  the  field- 
excitation, 

x"Q  =  synchronous   reactance   in   quadrature   with   the 

field  excitation, 
r  =  armature  resistance, 

i  =  armature  current, 

E  =  e\  -+-  je2  =  e(cos  6*  -f-  j  sin  6')  =  terminal  voltage, 
that  is, 

tan  0'  =  —  =  angle  of  lag  of  current  i  behind  terminal 

voltage,  e, 
the  nominal  generated  e.m.f.  of  the  machine  is 

(e,  +  n)J  +  (e2  +  zV)  (e,  +  z'V) 


(67) 


(e  cos  6'  +  ri)2  +  (e  sin  6'  -f  x'0i)  (e  sin  6'  +  x"0i) 
\/(e  cos  6'  +  ri)2  +  (e  sin  ^',  +  x"0i)2 


ARMATURE  REACTIONS  OF  ALTERNATORS     287 

and  the  field  excitation,  /0,  required  to  give  terminal  voltage,  6, 
at  current,  i,  and  angle  of  lag,  0',  is,  therefore 

e0          e0108  ^ 
Jo       a(P'n~27rfn*(S>' 

and  the  position  angle,  0,  between  the  field-excitation,  /0,  and 
the  armature  current,  i,  that  is,  between  the  direction  of  the 
field-poles  and  the  direction  in  which  the  armature  current 
reaches  its  maximum,  is 

e2  +  x"Qi       e  sin  0'  +  x"*i 

tan  6  —  t  —  -  :  —  r  =  --  ^7—  :  —  -•  (70) 

ei  +  n         e  cos  6'  +  n 

200.  At  non-inductive  load, 

61  =  e  and  e2  =  0  (71) 

from  (68), 

_  (e  +  r,)2  +  XoVV2 

'  V(e  +  ri)»  +  *Vi« 
If 

x'0  =  x"0  =  x0,  (73) 

that  is,  the  synchronous  reactance  of  the  machine  is  constant  in  all 
positions  of  the  armature,  or  in  other  words,  the  magnetic  per- 
meance, (P,  and  the  self  -inductive  reactance,  x$  do  not  vary  with 
the  position  of  the  armature  in  the  field,  equation  (68)  assumes 
the  form 


eo  =  V(ei  +  n)2  +  (e*  +  w)2,  (74) 

and  this  is  the  absolute  value  of  the  equation  (22) 

o  =  E  +  Z0I,  (22) 


derived  in  §195  for  the  case  of  uniform  synchronous  impedance. 
Substituting  in  (22), 

/  =  i,  and  E  =  e\  +  je2, 
and  expanding,  gives 

Eo  =  (ei  +  ,762)  +  i(r  +  jxo) 
=  (ei  +  ri)  +  j(e2  +  xQi)  ; 

thus,  the  absolute  value, 

(74) 


288         ALTERNATING-CURRENT  PHENOMENA 

201.  At  short-circuit,  and  approximately,  near  short-circuit, 
d  =  0  and  e2  =  0,  (75) 

equation  (68)  assumes  the  form 

o  . 


(76) 

v  T-  -r  XQ  - 

or  the  short-circuit  current, 


Since  x'Q  and  z"o  usually  are  large,  compared  with  r,  r  can  be 
neglected  in  equation  (77),  and  (77)  so  assumes  the  form 

to  =  %-,  (78) 

X  o 

that  is,  the  short-circuit  current  of  an  alternator, 

e0 

0  =  y7' 

x  o 

depends  only  upon  the  synchronous  reactance  of  the  armature 
in  the  direction  of  the  field-excitation,  x'o,  but  not  upon  the  syn- 
chronous reactance  of  the  armature  in  quadrature  position  to  the 
field-excitation,  X"Q. 

Near  open-circuit,  that  is,  in  the  range  where  the  machine 
regulates  approximately  for  constant  potential,  and  ix0  and  espe- 
cially ir  are  small  compared  with  e,  we  have,  for  non-inductive 
load,  from  equation  (72), 


(e  +  ri)* 
or,  approximately, 


hence,  expanded  by  the  binomial  series, 


ARMATURE  REACTIONS  OF  ALTERNATORS     289 

and,  dropping  terms  of  higher  order, 

.   x'oX0"i2       x0"H2 
e0  =  e  -f-  n  +  -         ---  ~  —  > 

6  A  6 

or 


.  oo 

60  =  6  +  n  H  ---  £—        "  ~  (79) 

For  x'o  =  £"o  =  x0,  this  equation  (79)  assumes  the  usual  form, 

e,  =  e  +  ri  +  ~  -•  (80) 

Z       6 

In  the  range  near  open-circuit,  for  non-inductive  load,  the 
regulation  of  the  machine  accordingly  depends  not  upon  the 
synchronous  reactance,  z'o,  nor  upon  z"0,  but  upon  the  equivalent 
synchronous  reactance, 


x'"0  =  Vx"0(2x'0  -x"0).  (81) 

That  is,  in  an  alternator  with  non-uniform  synchronous  re- 
actance, the  short-circuit  current  and  the  regulation  of  the 
machine  near  short-circuit,  depend  upon  the  value  of  the  syn- 
chronous reactance,  corresponding  to  the  position  of  the  arma- 
ture coils  parallel,  or  coaxial  with  the  field-poles,  z'o,  while  the 
regulation  of  the  machine  for  non-inductive  load,  in  the  range 
where  the  machine  tends  to  regulate  for  approximately  constant 
potential,  that  is,  near  open-circuit,  depends  upon  the  value  of 
the  synchronous  reactance,  X"'Q  =  VV'o(2x'0  —  x"o),  where  x'0 
and  X"Q  are  the  two  quadrature  components  of  the  synchronous 
reactance. 

That  is,  the  regulation  of  such  an  alternator  of  variable  syn- 
chronous reactance  cannot  be  calculated  from  open-circuit 
test  and  short-circuit  test,  or  from  the  magnetic  characteristic 
of  the  machine  at  open-circuit,  or  nominal  generated  e.m.f.,  and 
the  synchronous  reactance,  as  given  by  the  machine  at  short- 
circuit. 

For  instance,  if 

x'0  =  10  and  z"0  =  4, 

the  effective  synchronous  reactance  near  short-circuit, 

z'0  =  10; 
and  the  effective  synchronous  reactance  near  open-circuit, 


19 


290         ALTERNATING-CURRENT  PHENOMENA 

The  regulation  for  non-inductive  load  thus  is  better  than 
corresponds  to  the  short-circuit  impedance. 

From  equation  (68),  by  solving  for  the  terminal  voltage,  e, 
the  variation  of  the  terminal  voltage,  e,  with  change  of  load,  i, 
at  constant  field-excitation,  /0,  and  so  constant  nominal  gener- 
ated e.m.f.,  e0,  that  is,  the  regulation  curve  of  the  machine,  is 
calculated. 

For  instance,  for  non-inductive  load,  or  0'  =  0,  equation  (68), 
solved  for  e,  gives 


e  =  -  s'oz'V.H-  e,     ~  +  x0"*i*  (x"Q  -  x'*)*  -  ri.  (82) 

202.  As  illustrations  are  shown,  in  Fig.  142,  the  regulation 
curves,  with  the  terminal  voltage,  e,  as  ordinates,  and  the  cur- 
rent, i,  as  abscissas,  at  constant  field-excitation,  that  is,  constant 
nominal  generated  e.m.f.,  e0>  for  the  constants 

e0  =  2500  volts;  x'0   =  10  ohms; 

r  —         1  ohm;  x"Q  =    4  ohms; 

for  non-inductive  load         E  =  e,  (Curve  I.) 

and  for  inductive  load  of  60  per  cent,  power-factor,  E  =  e  (0  .  6  + 
0.8  j.)  (Curve  II.) 

For  comparison  are  plotted  in  the  same  figure,  in  dotted 
lines,  the  regulation  curves  for  constant  synchronous  reactance 

x0  =  10  ohms, 

that  is,  for  the  same  open-circuit  voltage  and  same  short-circuit 
current. 

As  seen  from  Fig.  142,  the  difference  between  the  two  regula- 
tion curves,  for  variable  and  for  constant  synchronous  reactance, 
is  quite  considerable  at  non-inductive  load,  but  practically  negli- 
gible at  highly  inductive  load.  This  is  to  be  expected,  since  at 
inductive  load  the  armature  current  reaches  its  maximum  nearly 
in  opposition  to  the  field-poles,  and  in  this  direction  the  syn- 
chronous reactance  is  the  same,  X'Q,  as  at  short-circuit. 

In  the  preceding  discussion  of  the  alternator  with  variable  syn- 
chronous reactance,  e.m.f.  and  current  are  assumed  as  sine 
waves.  The  periodic  variation  of  reactance  produces,  however, 
a  distortion  of  wave-shape,  consisting  mainly  of  a  third  harmonic 
which  superimposes  on  the  fundamental,  as  discussed  in  Chapter 
XXV.  The  preceding,  therefore,  applies  to  the  equivalent 


ARMATURE  REACTIONS  OF  ALTERNATORS    291 

sine  wave,  which  represents  approximately  the  actual  distorted 
wave. 

As  the  intensity,  and  the  phase  difference  between  the  third 
harmonic  and  the  fundamental  changes  with  the  load,  in  such 


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FIG.  142. 


an  alternator  of  pulsating  synchronous  reactance,  the  wave-shape 
of  the  machine  changes  more  or  less  with  the  load  and  the  char- 
acter of  the  load. 


CHAPTER  XXIII 
SYNCHRONIZING  ALTERNATORS 

203.  All  alternators,  when  brought  to  synchronism  with  each 
other,  operate  in  parallel  more  or  less  satisfactorily.     This  is  due 
to  the  reversibility  of  the  alternating-current  machine;  that  is, 
its  ability  to  operate  as  synchronous  motor.     In  consequence 
thereof,  if  the  driving  power  of  one  of  several  parallel-operating 
generators  is  withdrawn,  this  generator  will  keep  revolving  in 
synchronism  as  a  synchronous  motor;  and  the  power  with  which 
it  tends  to  remain  in  synchronism  is  the  maximum  power  which 
it  can  furnish  as  synchronous  motor  under  the  conditions  of 
running. 

204.  The  principal  and  foremost  condition  of  parallel  opera- 
tion of  alternators  is  equality  of  frequency;  that  is,  the  trans- 
mission of  power  from  the  prime  movers  to  the  alternators  must 
be  such  as  to  allow  them  to  run  at  the  same  frequency  without 
slippage  or  excessive  strains  on  the  belts  or  transmission  devices. 

Rigid  mechanical  connection  of  the  alternators  cannot  be  con- 
sidered as  synchronizing,  since  it  allows  no  flexibility  or  phase 
adjustment  between  the  alternators,  but  makes  them  essentially 
one  machine.  If  connected  in  parallel,  a  difference  in  the  field- 
excitation,  and  thus  the  generated  e.m.f.  of  the  machines,  may 
cause  large  cross-current,  since  it  cannot  be  taken  care  of  by 
phase  adjustment  of  the  machines. 

Thus  rigid  mechanical  connection  is  not  desirable  for  parallel 
operation  of  alternators. 

205.  The  second  important  condition  of  parallel  operation  is 
uniformity  of  speed;  that  is,  constancy  of  frequency.     If,  for 
instance,   two   alternators   are   driven   by  independent  single- 
cylinder  engines,  and  the  cranks  of  the  engines  happen  to  be 
crossed,  the  one  engine  will  pull,  while  the  other  is  near  the  dead- 
point,  and  conversely.     Consequently,  alternately  the  one  alter- 
nator will  tend  to  speed  up  and  the  other  slow  down,  then  the 
other  speed  up  and  the  first  slow  down.     This  effect,  if  not  taken 
care  of  by  fly-wheel  capacity,  causes  a  "hunting"  or  surging 

292 


SYNCHRONIZING  ALTERNATORS  293 

action ;  that  is,  a  fluctuation  of  the  voltage  with  the  period  of  the 
engine  revolution,  due  to  the  alternating  transfer  of  the  load 
from  one  engine  to  the  other,  which  may  even  become  so  excessive 
as  to  throw  the  machines  out  of  step,  especially  when  by  an  ap- 
proximate coincidence  of  the  period  of  engine  impulses  (or  a 
multiple  thereof),  with  the  natural  period  of  oscillation  of  the 
revolving  structure,  the  effect  is  made  cumulative.  This  diffi- 
culty as  a  rule  does  not  exist  with  turbine  or  water-wheel  driving, 
but  is  specially  severe  with  gas-engine  drive,  and  special  pre- 
cautions are  then  often  taken,  by  the  use  of  a  short-circuited 
squirrel  cage  winding  in  the  field  pole  faces. 

206.  In    synchronizing  alternators,   we  have  to  distinguish 
the   phenomena  taking  place  when  throwing  the  machines  in 
parallel  or  out  of  parallel,  and  the  phenomena  when  running  in 
synchronism. 

When  connecting  alternators  in  parallel,  they  are  first  brought 
approximately  to  the  same  frequency  and  same  voltage;  and  then, 
at  the  moment  of  approximate  equality  of  phase,  as  shown  by  a 
phase-lamp  or  other  device,  they  are  thrown  in  parallel. 

Equality  of  voltage  is  less  important  with  moderate  size  alter- 
nators than  equality  of  frequency,  and  perfect  equality  of  phase  is 
usually  of  importance  only  in  avoiding  an  instantaneous  flickering 
of  the  light  of  lamps  connected  to  the  system.  When  two  alter- 
nators are  thrown  together,  currents  exist  between  the  machines, 
which  accelerate  the  one  and  retard  the  other  machine  until 
equal  frequency  and  proper  phase  relation  are  reached. 

With  modern  ironclad  alternators,  this  interchange  of  mechan- 
ical power  is  usually,  even  without  very  careful  adjustment  before 
synchronizing,  sufficiently  limited  not  to  endanger  the  machines 
mechanically,  since  the  cross-currents,  and  thus  the  interchange 
of  power,  are  limited  by  self-induction  and  armature  reaction. 

In  machines  of  very  low  armature-reaction,  that  is,  machines 
of  "very  good  constant-potential  regulation,"  much  greater  care 
has  to  be  exerted  in  the  adjustment  to  equality  of  frequency, 
voltage,  and  phase,  or  the  interchange  of  current  may  become 
so  large  as  to  destroy  the  machine  by  the  mechanical  shock;  and 
sometimes  the  machines  are  so  sensitive  in  this  respect  that  it 
is.  difficult  to  operate  them  in  parallel.  The  same  applies  in 
getting  out  of  step. 

207.  When  running  in  synchronism,  nearly  all  types  of  ma- 
chines will  operate  satisfactorily;  a  medium  amount  of  armature 


294         AL  TERN  A  TING-C  URRENT  PHENOMENA 


reaction  is  preferable,  however,  such  as  is  given  by  modern  alter- 
nators—not too  high  to  reduce  the  synchronizing  power  too 
much,  nor  too  low  to  make  the  machine  unsafe  in  case  of  accident, 
such  as  falling  out  of  step,  etc. 

If  the  armature  reaction  is  very  low,  an  accident — such  as  a 
short-circuit,  falling  out  of  step,  opening  of  the  field  circuit,  etc. 
— may  destroy  the  machine.  If  the  armature  reaction  is  very 
high,  the  driving  power  has  to  be  adjusted  very  carefully  to 
constancy,  since  the  synchronizing  power  of  the  alternators  is  too 
weak  to  hold  them  in  step  and  carry  them  over  irregularities  of 
the  driving-power. 

208.  Series  operation  of  alternators  is  possible  only  by  rigid 
mechanical  connection,  or  by  some  means  whereby  the  machines, 
with  regard  to  their  synchronizing  power,  act  essentially  in  par- 


FIG.  143. 

allel;  as,  for  instance,  by  the  arrangement  shown  in  Fig.  143, 
where  the  two  alternators,  AI,  Az,  are  connected  in  series,  but 
interlinked  by  the  two  coils  of  a  transformer,  Ty  of  which  the  one 
is  connected  across  the  terminals  of  one  alternator  and  the  other 
across  the  terminals  of  the  other  alternator  in  such  a  way  that, 
when  operating  in  series,  the  coils  of  the  transformer  will  be  with- 
out current.  In  this  case,  by  interchange  of  power  through  the 
transformers,  the  series  connection  will  be  maintained  stable. 

209.  In  two  parallel  operating  alternators,  as  shown  in  Fig. 
144,  let  the  voltage  at  the  common  busbars  be  assumed  as  zero 
line,  or  real  axis  of  coordinates  of  the  complex  representation; 
and  let 

e    =  difference  of  potential  at  the  common  busbars  of  the 
two  alternators; 


SYNCHRONIZING  ALTERNATORS 


295 


Z  =  r  +  jx  =  impedance  of  the  external  circuit; 
Y  =  g  —  jb  =  admittance  of  the  external  circuit; 

hence,  the  current  in  the  external  circuit  is 

e 


Let 


r  +  jx 


=  e(g  - 


generated  e.m.f.   of 


Ei  =  ei  +  je\  =  ai(cos  0i  +  j  sin  Oi) 

first  machine; 
E2  =  e2  +  je'z  =  a2(cos  62  +  j  sin  02)    =   generated  e.m.f.   of 

second  machine; 
/i   =  ii  —  ji\  =  current  of  the  first  machine; 

1  2   =  it  —  jif2  =  current  of  the  second  machine; 

Zi  =  TI  +  jxi  =  internal  impedance,  and  FI  =  g\  —  jbi  =  inter- 

nal admittance  of  the  first  machine; 
Zz  =  r2  +  jx2  =  internal  impedance,  and  F2  =  gz  —  jb2  =  inter- 

nal admittance  of  the  second  machine. 


Then, 


FIG.  144. 


+  e'S  =  a!2; 


=  a22; 


i  =  e  +  I\Zi,  or  d  +  je'i  =  (e  +  iiri  +  i\Xi)  + 
*  =  e  +  IzZ2,  or  e2  -f  je'z  =  (e  +  izrz  +  ^2^2 
=  /i  +  ^2,  or  e^  -  jeb     =  (*i  +  «2)  -  j(i' 

This  gives  the  equations: 

61  =  e  +  i>i  +  i'\Xi\ 
ez  =  e  +  2>2  +  t^^ 


296         AL  TERN  A  TING-C  URRENT  PHENOMENA 
e'i 


=  i2x2  —  i'2r2', 


eg  =  ii  +  iz'j 
eb  =  i\  +  i'2 


or  eight  equations  with  nine  variables,  e\,  e'i,  e2,  e'2,  ii,  i'i,  i2,  i'2,  e. 
Combining  these  equations  by  twos, 

eiri  +  e'&i  =  eri  +  itfi2; 
e2r2  +  e'2x2  =  er2  +  i2z22; 
substituting  in 

ii  +  iz  =  eg, 
we  have 

ei^i  +  e'ibi  +  e2^2  +  e'2&2  =  e(gi  +  ^2  -f  g); 

and  analogously, 


e(bi  +  62  +  6)  : 
dividing, 


e262  -  e'lgfi  -  e'2^2' 
substituting 

g  =  y  cos  a  d   =  ai  cos  0i        e2   =  a2  cos  02 

6  =  i/  sin  a  e'\  =  ai  sin  61        e'2  =  a2  sin  02 

gives 

0  +  ^1  +  ^2  _  aij/i  cos  («i  —  0i)  +  a2y2  cos  («2  — 


6  +  61  +  62       aiyi  sin  (a\  —  61)  +  azy2  sin  (a2  —  02) 

as  the  equation  between  the  phase  displacement  angles,  0i  and  02, 
in  parallel  operation. 

The  power  supplied  to  the  external  circuit  is, 

P  =  e2g, 

of  which  that  supplied  by  the  first  machine  is, 

PI  —  eii; 

by  the  second  machine, 

p2  =  eiz. 

The  total  electrical  power  of  both  machines  is, 
P  =  Pl  +  P2, 


SYNCHRONIZING  ALTERNATORS  297 

of  which  that  of  the  first  machine  is, 

P  °          '  */ 

and  that  of  the  second  machine, 

The  difference  of  output  of  the  two  machines  is, 

AP  T=  Pi  -  P*  =  e  (ii  -  i2) ; 
denoting 

/}          |       n  n  n 

VI    ~[~    "2     _  "l     C72     _ 

~T~         ~T~ 

—  may  be  called  the  synchronizing  power  of  the  machines, 

or  the  power  which  is  transferred  from  one  machine  to  the  other 
by  a  change  of  the  relative  phase  angle. 

210.  SPECIAL  CASE. — Two  equal  alternators  of  equal  excitation. 

a\  =  a2  =  a, 

ZF7  f7 

1     ^~     A/  2     =~=     *^Q* 

Substituting  this  in  the  eight  initial  equations,  these  assume 
the  form, 

ei-  =  e  +  iir0  +  I'M, 

62     =  6  +  ItfQ  +  1  2^0» 


eg  =  ii  +  iz, 

eb  =  K  +  ,'',, 

ei2  +  e/2  =  622  +  62'2  =  a2. 

Combining  these  equations  by  twos, 

61  +  62  =  2  e  +  e  (rQg  +  xQb), 

e'i  +  e'2  =  e  (xQg  —  rQb) ; 
substituting 

61  =  a  cos  0i, 
e'i  =  a  sin  0i, 

62  =  a  cos  02, 
e'2  =  a  sin  02, 

we  have 

a  (cos  0i  +  cos  02)  =  e  (2  +  r0g  +  z 
a  (sin  0i  +  sin  02)  =  e  (xQg  -  r06) ; 


298         ALTERNA  TING-C URRENT  PHENOMENA 

expanding  and  substituting 

6 
e  =  - 


02 


5  = 


2 


gives 


-    , 
a  cos  e  cos  5  =  e  (  1  H 


/  -    ,   r0g  +Xo\ 
=  e  (  1  H  ---  •=  --  j 


a  sin  e  cos  5  =  e 


x0g  —  r06. 


hence 

That  is, 
and 

or, 


tan  «  = 


=  constant. 


1  +  62  =  constant; 


a  cos  5 


e  = 


-  rQb\  2 


at  no-phase  displacement  between  the  alternators,  or, 


we  have 


+  a?ob.  2 


-  r06\  2. 


From  the  eight  initial  equations  we  get,  by  combination, 

eirQ  +  e'lXo  =  e0rQ 
e2r0  +  e'2xQ  =  eQrQ 

subtracted  and  expanded, 

TO  (ei  —  ez)  +  xQ  (e'i  —  c'j)  . 


or,  since 


ex  —  e2  =  a  (cos  0i  —  cos  02)  =  —  2  a  sin  e  sin  5, 
'j  —  e'z  =  a  (sin  0i  —  sin  02)  =        2  a  cos  c  sin  5, 


SYNCHRONIZING  ALTERNATORS  299 

we  have 

2  a  sin  8  . 
ii  —  ?2  =  — • — 2 —  \XQ  cos  e  ~  r°  sm  cl 

£() 

=  2  ai/o  sin  d  sin  (a  —  c), 
where 

X0 

tan  a  =  —  • 

7*0 

The  difference  of  output  of  the  two  alternators  is 
AP  =  Pi  -P2  =  e(ii  -  *2); 

hence,  substituting, 

2ae  sin  8  ,  .      } 

AP  =    -     — * —     \XQ  cos  e  —  r0  sm  ej ; 
Zo 

substituting, 

a  cos  8 


e  = 


/A    ,   rpflf  +  x06\  2      /xo^f  -rp6\  2 
VI1          ~2       /   •"  V        2        / 


sm  e  =• 


VI 


cos  e  = 


+  r*g_x^   +  (M_!^ 
we  have, 

2a2sin  6  cos  8  j  x0  (l  +  — — ^r~~)   —  ro 
AP  =  


expanding, 

a2  sin  2  5 


or 

a2  sin  2  6 
AP  = 


2/o2 


A5 


300         ALTERNATING-CURRENT  PHENOMENA 

Hence,  the  transfer  of  power  between  the  alternators,  AP,  is  a 
maximum,  if  6  =  45°;  or  0i  —  02  =  90°;  that  is,  when  the  alter- 
nators are  in  quadrature. 

The  synchronizing  power,  — ,  is  a  maximum  if  6  =0;  that  is, 

the  alternators  are  in  phase  with  each  other. 
211.  As  an  instance,  curves  may  be  plotted  for, 

a  =  2500, 
Z0  =  rQ  +  jx0  =  1  +  10  j]  or  F0  =  go  -  jb0  =  0.01  -  0.1  j, 

f\ n 

with  the  angle,  5  =  — ~ — ,  as  abscissas,  giving 

40 

the  value  of  terminal  voltage,  e\ 

the  value  of  current  in  the  external  circuit,  ?  =  ey, 

the  value  of  interchange  of  current  between  the   alternators 

i\  —  iz] 
the  value  of  interchange  of  power  between  the  alternators,  AP 

=  Pi  -  P2; 

the  value  of  synchronizing  power,  — • 
For  the  condition  of  external  circuit, 

0  =  0,  6  =  0,  y  =  0, 

0.05,  0,  0.05, 

0.08,  0,  0.08, 

0.03,  +0.04,  0.05, 

0.03,  -0.04,  0.05. 


CHAPTER  XXIV 

SYNCHRONOUS  MOTOR 

212.  In  the  chapter  on  synchronizing  alternators  we  have  seen 
that  when  an  alternator  running  in  synchronism  is  connected  with 
a  system  of  given  voltage,  the  work  done  by  the  alternator  can  be 
either  positive  or  negative.  In  the  latter  case  the  alternator 
consumes  electrical,  and  consequently  produces  mechanical, 
power;  that  is,  runs  as  a  synchronous  motor,  so  that  the  investi- 
gation of  the  synchronous  motor  is  already  contained  essentially 
in  the  equations  of  parallel-running  alternators. 

Since  in  the  foregoing  we  have  made  use  mostly  of  the  sym- 
bolic method,  we  may  in  the  following,  as  an  example  of  the 
graphical  method,  treat  the  action  of  the  synchronous  motor 
graphically. 

Let  an  alternator  of  the  e.m.f.,  Ei,  be  connected  as  synchron- 
ous motor  with  a  supply  circuit  of  e.m.f.,  EQ,  by  a  circuit  of  the 
impedance,  Z. 

If  EQ  is  the  e.m.f.  impressed  upon  the  motor  terminals,  Z  is 
the  impedance  of  the  motor  of  generated  e.m.f.,  EI.  If  EQ  is  the 
e.m.f.  at  the  generator  terminals,  Z  is  the  impedance  of  motor  and 
line,  including  transformers  and  other  intermediate  apparatus. 
If  EQ  is  the  generated  e.m.f.  of  the  generator,  Z  is  the  sum  of  the 
impedances  of  motor,  line,  and  generator,  and  thus  we  have  the 
problem,  generator  of  generated  e.m.f.,  EQ,  and  motor  of  generated 
e.m.f.,  EI;  or,  more  general,  two  alternators  of  generated  e.m.fs., 
EQ,  EI,  connected  together  into  a  circuit  of  total  impedance,  Z. 

Since  in  this  case  several  e.m.fs.  are  acting  in  circuit  with  the 
same  current,  it  is  convenient  to  use  the  current,  I,  as  zero  line 
01  of  the  polar  diagram.  (Fig.  145.) 

If  /  =  i  =  current,  and  Z  =  impedance,  r  =  effective  resist- 
ance, x  =  effective  reactance,  and  z  =  V>2  +  xz  =  absolute 
value  of  impedance,  then  the  e.m.f.  consumed  by  the  resistance 
is  EH  —  ri,  and  is  in  phase  with  the  current;  hence  represented 
by  vector  OEu',  and  the  e.m.f.  consumed  by  the  reactance  is 
Ez  =  xi,  and  90°  ahead  of  the  current ;  hence  the  e.m.f.  consumed 

301 


302         AL  TERN  A  TING-C  URREN  T  PHENOMENA 


by  the  impedance  is  E  =  \/(^n)2  +  (Ez)2,  or  = 

and  ahead  of  the  current  by  the  angle  5,  where  tan  5  =  -. 

We  have  now  acting  in  circuit  the  e.m.fs.,  E,  EI,  E0;  or  EI  and 
E  are  components  of  EQ,  that  is,  E0  is  the  diagonal  of  a  parallelo- 
gram, with  EI  and  E  as  sides. 

Since  the  e.m.fs.  EI,  E0,  E,  are  represented  in  the  diagram, 
Fig.  145,  by  the  vectors  OE1}  OE0,  OE,  to  get  the  parallelogram 
of  EQ,  Eif  E,  we  draw  arcs  of  circles  around  0  with  EQ,  and  around 
E  with  EI.  Their  point  of  intersection  gives  the  impressed  e.m.f., 
OEp  =  EQ,  and  completing  the  parallelogram,  OEEQEi,  we  get, 
OEi  =  Eif  the  generated  e.m.f.  of  the  motor. 

<  IOEo  is  the  difference  of  phase  between  current  and  impressed 

e.m.f.,  or  generated  e.m.f.  of  the  generator. 

<  IOEi  is  the  difference  of  phase  between  current  and  generated 

e.m.f.  of  the  motor. 

And  the  power  is  the  current,  i,  times  the  projection  of  the  e.m.f. 
upon  the  current,  or  the  zero  line,  01. 

Hence,  dropping  perpendiculars,  EQEQl  and  EiEi1,  from  E0  and 
EI  upon  01,  it  is — 

Po  =  i  X  OEo1  =  power   supplied   by  generator  e.m.f.  of   gen- 
erator; 

PI  =  2  X  OEi1  =  electric   power  transformed   into   mechanical 
power  by  the  motor; 

p    ==  {  x  OEu  =  power  consumed  in  the  circuit  by  effective 

resistance. 
Obviously  P0  =  Pi  +  P. 

Since  the  circles  drawn  with  EQ  and  EI  around  0  and  E,  re- 
spectively, intersect  twice,  two  diagrams  exist.  In  general,  in 
one  of  these  diagrams  shown  in  Fig.  145  in  full  lines,  current 
and  e.m.f.  are  in  the  same  direction,  representing  mechanical 
work  done  by  the  rnachine  as  motor.  In  the  other,  shown  in 
dotted  lines,  current  and  e.m.f.  are  in  opposite  direction,  repre- 
senting mechanical  work  consumed  by  the  machine  as  generator. 

Under  certain  conditions,  however,  EQ  is  in  the  same,  E\  in 
opposite  direction,  with  the  current;  that  is,  both  machines  are 
generators. 

213.  It  is  seen  that  in  these  diagrams  the  e.m.fs.  are  considered 
from  the  point  of  view  of  the  motor;  that  is,  work  done  as  syn- 
chronous motor  is  considered  as  positive,  work  done  as  generator 


SYNCHRONOUS  MOTOR 


303 


is  negative.     In  the  chapter  on  synchronizing  generators  we 
took  the  opposite  view,  from  the  generator  side. 

In  a  single  unit-power  transmission,  that  is,  one  generator 
supplying  one  synchronous  motor  over  a  line,  the  e.m.f.  con- 
sumed by  the  impedance,  E  =  OE,  Figs.  146  to  148,  consists 
three  components;  the  e.m.f.,  OE%1  —  E%,  consumed  by  the  im- 
pedance of  the  motor,  the  e.m.f.,  Ez^Es1  =  E$  consumed  by  the 
impedance  of  the  line,  and  the  e.m.f.,  E^E  =  E*,  consumed  by 


FIG.  145. 

the  impedance  of  the  generator.  Hence,  dividing  the  opposite 
side  of  the  parallelogram,  EiEo,  in  the  same  way,  we  have:  OEi  — 
E\  —  generated  e.m.f.  of  the  motor,  OEz  =  E%  =  e.m.f.  at  motor 
terminals  or  at  end  of  line,  OE$  =  Es  =  e.m.f.  at  generator 
terminals,  or  at  beginning  of  line.  OEQ  =  EQ  =  generated  e.m.f. 
of  generator. 

The  phase  relation  of  the  current  with  the  e.m.fs.,  Ei,Eot  de- 
pends upon  the  current  strength  and  the  e.m.fs.,  EI  and  EQ. 

214.  Figs.  146  to  148  show  several  such  diagrams  for  different 
values  of  EI,  but  the  same  value  of  I  and  EQ.  The  motor  diagram 
being  given  in  drawn  line,  the  generator  diagram  in  dotted  line. 

As  seen,  for  small  values  of  EI  the  potential  drops  in  the  alter- 
nator and  in  the  line.  For  the  value  of  EI  =  EQ  the  potential 
rises  in  the  generator,  drops  in  the  line,  and  rises  again  in  the 


304         ALTERNATING-CURRENT  PHENOMENA 


FIG.  146. 


FIG.  147. 


SYNCHRONOUS  MOTOR 


305 


FIG.  148. 


6, 


FIG.  149. 


20 


306         ALTERNATING-CURRENT  PHENOMENA 

motor.  For  larger  values  of  EI,  the  potential  rises  in  the  alter- 
nator as  well  as  in  the  line,  so  that  the  highest  potential  is  the 
generated  e.m.f.  of  the  motor,  the  lowest  potential  the  generated 
e.m.f.  of  the  generator. 

It  is  of  interest  now  to  investigate  how  the  values  of  these 
quantities  change  with  a  change  of  the  constants. 

215.  A.  Constant  impressed  e.m.f.,  EQ,  constant-current  strength 
I  =  i,  variable  motor  excitation,  E\.  (Fig.  149.) 

If  the  current  is  constant,  =  i;  OE,  the  e.m.f.  consumed  by 
the  impedance,  and  therefore  point,  Ej  are  constant.  Since  the 
intensity,  but  not  the  phase  of  EQ  is  constant,  EQ  lies  on  a  circle 
e0  with  EQ  as  radius.  From  the  parallelogram,  OEEoEi  follows, 
since  EiEQ  parallel  and  =  OE,  that  E\  lies  on  a  circle,  ei,  con- 
gruent to  the  circle,  eQ,  but  with  Ei,  the  image  of  E,  as  center; 
OEi  =  OE. 

We  can  construct  now  the  variation  of  the  diagram  with  the  va- 
riation of  Ei',  in  the  parallelogram,  OEE0Eij  0,  and  E  are  fixed, 
and  EQ  and  EI  move  on  the  circles,  e0  and  e\,  so  that  E0Ei  is 
parallel  to  OE. 

The  smallest  value  of  EI  consistent  with  current  strength,  I,  is 
Oli  =  Ei,  01  =  EQ.  In  this  case  the  power  of  the  motor  is 
Oli1  X  /,  hence  already  considerable.  ^Increasing  EI  to  02i,  03i, 
etc.,  the  impressed  e.m.fs.  move  to  02,  03,  etc.,  the  power  is  I  X 
0211,  I  X  OSi1,  etc.,  increases  first,  reaches  the  maximum  at  the 
point  3i,  3,  the  most  extreme  point  at  the  right,  with  the  im- 
pressed e.m.f.  in  phase  with  the  current,  and  then  decreases 
again,  while  the  generated  e.m.f.  of  the  motor,  E\,  increases  and 
becomes  =  EQ  at  4i,  4.  At  5i,  5,  the  power  becomes  zero,  and 
further  on  negative;  that  is,  the  motor  has  changed  to  a  generator, 
and  produces  electrical  energy,  while  the  impressed  e.m.f.,  e0, 
still  furnishes  electrical  energy — that  is,  both  machines  as  gen- 
erators feed  into  the  line,  until  at  61,  6,  the  power  of  the  impressed 
e.m.f.,  EQ,  becomes  zero,  and  further  on  energy  begins  to  flow 
back;  that  is,  the  motor  is  changed  to  a  generator  and  the  genera- 
tor to  a  motor,  and  we  are  on  the  generator  side  of  the  diagram. 
At  7i,  7,  the  maximum  value  of  EI,  consistent  with  the  current, 
/,  has  been  reached,  and  passing  still  further  the  e.m.f.,  E\  de- 
creases again,  while  the  power  still  increases  up  to  the  maximum 
at  81,  8,  and  then  decreases  again,  but  still  E\  remaining  generator, 
EQ  motor,  until  at  Hi,  11,  the  power  of  EQ  becomes  zero;  that  is, 
EQ  changes  again  to  a  generator,  and  both  machines  are  generators, 


SYNCHRONOUS  MOTOR  307 

up  to  12i,  12,  where  the  power  of  EI  is  zero,  E\,  changes  from 
generator  to  motor,  and  we  come  again  to  the  motor  side  of  the 
diagram,  and  the  power  of  the  motor  increases  while  E\  still 
decreases,  until  li,  1,  is  reached. 

Hence,  there  are  two  regions,  for  very  large  EI  from  5  to  6  and 
for  very  small  EI  from  11  to  12,  where  both  machines  are  genera- 
tors; otherwise  the  one  is  generator,  the  other  motor. 

For  small  values  of  EI  the  current  is  lagging,  begins,  however, 
at  2  to  lead  the  generated  e.m.f.  of  the  motor,  EI,  at  3  the  gener- 
ated e.m.f.  of  the  generator,  EQ. 

It  is  of  interest  to  note  that  at  the  smallest  possible  value  of 
EI,  li,  the  power  is  already  considerable.  Hence,  the  motor 
can  run  under  these  conditions  only  at  a  certain  load.  If  this 
load  is  thrown  off,  the  motor  cannot  run  with  the  same  current, 
but  the  current  must  increase.  We  have  here  the  curious  con- 
dition that  loading  the  motor  reduces,  unloading  increases,  the 
current  within  the  range  between  1  and  12. 

The  condition  of  maximum  output  is  3,  current  in  phase  with 
impressed  e.m.f.  Since  at  constant  current  the  loss  is  constant, 
this  is  at  the  same  time  the  condition  of  maximum  efficiency;  no 
displacement  of  phase  of  the  impressed  e.m.f.,  or  self-induction 
of  the  circuit  compensated  by  the  effect  of  the  lead  of  the  motor 
current.  This  condition  of  maximum  efficiency  of  a  circuit  we 
have  found  already  in  Chapter  XI. 

216.  B.     EQ  and  EI  constant,  I  variable. 

Obviously  EQ  lies  again  on  the  circle  e0  with  EQ  as  radius  and 
0  as  center. 

E  lies  on  a  straight  line,  e,  passing  through  the  origin. 

Since  in  the  parallelogram,  OEE0Ei,  EEQ  =  EI,  we  derive  EQ 
by  laying  a  line,  EEo  =  EI,  from  any  point,  E,  in  the  circle,  eQ, 
and  complete  the  parallelogram. 

All  these  lines,  EEo,  envelop  a  certain  curve,  ei,  which  can  be 
considered  as  the  characteristic  curve  of  this  problem,  just  as 
circle,  ei,  in  the  former  problem. 

These  curves  are  drawn  in  Figs.  150,  151,  152,  for  the  three 
cases:  1st,  Ei  =  EQ',  2d,  Ei<E0;  3d,  Ei>EQ. 

In  the  first  case,  EI  =  EQ  (Fig.  150),  we  see  that  at  very  small 
current,  that  is  very  small  OE,  the  current,  I,  leads  the  impressed 
e.m.f.,  EQ,  by  an  angle,  E1QOI  =  00.  This  lead  decreases  with 
increasing  current,  becomes  zero,  and  afterward  for  larger  cur- 
rent, the  current  lags.  Taking  now  any  pair  of  corresponding 


308         ALTERNATING-CURRENT  PHENOMENA 

points,  E,  EQ,  and  producing  EE0  untiHt  intersects  ei}  in 
have  <  E&E  =  90°,  #1  =  E0)  thus:  OEi  =  ##0  =  OF0  = 


we 


FIG.  150. 


FIG.  151. 

that  is,  EEi  =  2  E0. That  means  the  characteristic  curve,  d,  is 

the  envelope  of  lines  EEi,  of  constant  lengths,  2  #o,  sliding  between 
the  legs  of  the  right  angle,  E1OE'}  hence,  it  is  the  sextic  hypocy- 


SYNCHRONOUS  MOTOR 


309 


cloid  osculating  circle,  €Q,  which  has  the  general  equation,  with 
e,  6i  as  axes  of  coordinates, 


In  the  next  case,  EI  <  EQ  (Fig.  151),  we  see  first,  that  the 
current  can  never  become  zero  like  in  the  first  case,  EI  =  EQ, 
but  has  a  minimum  value  corresponding  to  the  minimum  value 

r  TTFr/        T/  -^0   —  •"!  i  •  i  Tit  EQ  —  EI 

of  OE  :  1  1  =  -  ,  and  a  maximum  value:  I  i  =  --- 

2  z 

Furthermore,  the  current  may  never  lead  the  impressed  e.m.f., 
EQ,  but  always  lags.  The  minimum  lag  is  at  the  point,  H.  The 
locus,  61,  as  envelope  of  the  lines,  EE0,  is  a  finite  sextic  curve, 
shown  in  Fig.  151. 


FIG.  152. 

If  EI  <  EQ,  at  small  EQ  —  EI,  H  can  be  below  the  zero  line, 
and  a  range  of  leading  current  exists  between  two  ranges  of  lag- 
ging currents. 

In  the  case,  E\  >  EQ  (Fig.  152),  the  current  cannot  equal  zero 

either,  but  begins  at  a  finite  value,  I\,  corresponding  to  the  mini- 

•pi   -pi 

mum  value  of  OE,  lf\  —  -          —-At  this  value,  however,  the 

alternator,  EI,  is  still  generator  and  changes  to  a  motor,  its  power 
passing  through  zero,  at  the  point  corresponding  to  the  vertical 
tangent,  upon  e\,  with  a  very  large  lead  of  the  impressed  e.m.f. 
against  the  current.  At  H  the  lead  changes  to  lag. 


310 


AL  TERN  A  TING-C  URREN  T  PHENOMENA 


The  minimum  and  maximum  values  of  current  in  the  three 
conditions  are  given  by: 

Maximum 

2E0 


Minimum 
1st.    7  =  0, 

EQ  — 


2d.    /  = 


3d.    I  = 


z 

El    —    EQ 


I    = 
I    = 

I    = 


z 
EQ  + 


Since  the  current  in  the  line  at  EI  =  O,  that  is,  when  the  motor 

Tjl 

stands  still,  is  70  =  — ,  we  see  that  in  such  a  synchronous  motor- 
plant,  when  running  at  synchronism,  the  current  can  rise  far  be- 
yond the  value  it  has  at  standstill  of  the  motor,  to  twice  this 
value  at  1,  somewhat  less  at  2,  but  more  at  3. 


FIG.  153. 

217.  C.  EQ  =  constantj  EI  varied  so  that  the  efficiency  is  a 
maximum  for  all  currents.  (Fig.  153.) 

Since  we  have  seen  that  the  output  at  a  given  current  strength, 
that  is,  a  given  loss,  is  a  maximum,  and  therefore  the  efficiency 
a  maximum,  when  the  current  is  in  phase  with  the  generated 
e.m.f.,  EQ)  of  the  generator,  we  have  as  the  locus  of  EQ  the  point, 
EQ  (Fig.  153),  and  when  E  with  increasing  current  varies  on  e, 
E\  must  vary  on  the  straight  line,  e\,  parallel  to  e. 


SYNCHRONOUS  MOTOR  311 

Hence,  at  no-load  or  zero  current,  EI  =  EQ,  decreases  with 
increasing  load,  reaches  a  minimum  at  OEi1  perpendicular  to  e\, 
and  then  increases  again,  reaches  once  more  EI  =  EQ  at  Ei2,  and 
then  increases  beyond  EQ.  The  current  is  always  ahead  of  the 
generated  e.m.f.,  EI,  of  the  motor,  and  by  its  lead  compensates 
for  the  self-induction  of  the  system,  making  the  total  circuit  non- 
inductive. 

The  power  is  a  maximumjat  Ei3,  where  OEi4  =  Ei*E0  =  0.5  X 

OEo,  and  is  then  =  I  X  §•     Since  OES  =  Ir  =  ~,  I  =  ~ 

±j  &  2i  T 

E  2 
and  P  =  ~-,  hence  =  the  maximum  power  which,  over  a  non- 

inductive  line  of  resistance  r  can  be  transmitted,  at  50  per  cent. 
efficiency,  into  a  non-inductive  circuit. 
In  this  case, 

z       EQ 


In  general,  it  is,  taken  from  the  diagram,  at  the  condition  of 
maximum  efficiency, 


Comparing  these  results  with  those  in  Chapter  XI  on  Induct- 
ive and  Condensive  Reactance,  we  see  that  the  condition  of 
maximum  efficiency  of  the  synchronous  motor  system  is  the  same 
as  in  a  system  containing  resistance  and  condensive  reactance, 
fed  over  an  inductive  line,  the  lead  of  the  current  against  the 
generated  e.m.f.,  EI,  here  acting  in  the  same  way  as  the  con- 
denser capacity  in  Chapter  XI. 

218.  D.  EQ  =  constant;  PI  =  constant. 

If  the  power  of  a  synchronous  motor  remains  constant,  we 
have  (Fig.  154)  I  X  OEi1  =  constant,  or,  since  OE1  =  Ir,  I  = 


and  OE1  X  OES  =  OE1  X  ElE0l  =  constant. 


Hence  we  get  the  diagram  for  any  value  of  the  current,  I,  at 
constant  power,  PI,  by  making  OE1  =  Ir,  E1EQ1  =  ~j  erecting 

in  EQl  a  perpendicular,  which  gives  two  points  of  intersection 
with  circle,  eQ,  EQ,  one  leading,  the  other  lagging.  Hence,  at  a 
given  impressed  e.m.f.,  EQ,  the  same  power,  PI,  can  be  trans- 
mitted by  the  same  current,  7,  with  two  different  generated 
e.m.fs.,  EI,  of  the  motor;  one,  OEi  =  EEQ  small,  corresponding 


312         ALTERNATING-CURRENT  PHENOMENA  t 

to  a  lagging  current;  and  the  other,  OEi  =  EE0  large,  corre- 
sponding to  a  leading  current.  The  former  is  shown  in  dotted 
lines,  the  latter  in  full  lines,  in  the  diagram,  Fig.  154. 

Hence  a  synchronous  motor  can  work  with  a  given  output,  at 
the  same  current  with  two  different  counter  e.m.fs.,  E\.  In  one 
of  the  cases  the  current  is  leading,  in  the  other  lagging. 


FIG.  154. 

In  Figs.  155  to  158  are  shown  diagrams,  giving  the  points 

EQ  =  impressed  e.m.f.,  assumed  as  constant  =  1000  volts, 
E    =  e.m.f.  consumed  by  impedance,  '••-,, 

El  —  e.m.f.  consumed  by  resistance  (not  numbered). 
The  counter  e.m.f.  of  the  motor,  Ei,  is  OE\,  equal  and  parallel 
EE0,  but  not  shown  in  the  diagrams,  to  avoid  complication. 

The  four  diagrams  correspond  to  the  values  of  power,  or  motor 
output, 

P  =     1,000,  6,000,     9,000,     12,000  watts,  and  give: 

P  =    1,000       46  <  Ei  <  2,200,  1  <  /  <  49  Fig.  155. 

P  =    6,000  340  <  El  <  1,920,          7  <  /  <  43  Fig.  156. 

P  =    9,000  540  <  El  <  1,750,      11.8  <  7  <  38.2  Fig.  157. 

P  =  12,000  920  <  El  <  1,320,     20     <  /  <  30  Fig.  158. 

As  seen,  the  permissible  value  of  counter  e.m.f.,  Ei,  and  of 
current,  /,  becomes  narrower  with  increasing  output. 


SYNCHRONOUS  MOTOR 


313 


E0=1000 
P=1000 
46<E'i<'2200 


2170 
2120 


45.5 

40 
37.5 


1050/1840  2/25 

1480  32 

1100  /1580  31/16.7 

1250  7 


FIG.  155. 


E0=1000 
P=6000 
340<Ei<1920 
<  43 


El  I 

340  17.3 

430/630  10/30 

750/1090  8/37.5 


900/1720       7/43 


10.40/1920      8/37.5 

1170/1810       10/30 
1450  17.3 


FIG.  156. 


314 


ALTERNATING-CURRENT  PHENOMENA 


In  the  diagrams,  different  points  of  EQ  are  marked  with  1,  2, 
3  .  .  .,  when  corresponding  to  leading  current,  with  21,  31, 
.  .  . ,  when  corresponding  to  lagging  current. 


Eo=1000 

P=9000 

540<E,<1750 

11.8<I<38.2 


540  21.2 

620/820      15/30 
720/1100     13/34.7 


900/1590     11.8/38.2^ 


1080/1750     13/34.7 
1200/1660    15/30 
1440  21.2 


FIG.  157. 


E0=1000 

P=12000 

920<E,<1320 

20<I<30 


El  I 

3'      920  24.5 

2'     920/1100  21/28.6 

1000/1260  20/30 

2      1120/1320  21/28.6 


1280 


24.5 


FIG.  158. 

The  values  of  counter  e.m.f.,  E\,  and  of  current,  /,  are  noted 
on  the  diagrams,  opposite  to  the  corresponding  points,  EQ. 


SYNCHRONOUS  MOTOR  315 

In  this  condition  it  is  interesting  to  plot  the  current  as  function 
of  the  generated  e.m.f.,  Ei,  of  the  motor,  for  constant  power,  PI. 
Such  curves  are  given  in  Fig.  162  and  explained  in  the  following 
on  page  430. 

219.  While  the  graphic  method  is  very  convenient  to  get  a 
clear  insight  into  the  interdependence  of  the  different  quantities, 
for  numerical  calculation  it  is  preferable  to  express  the  diagrams 
analytically. 

For  this  purpose, 

Let  z  =  A/r2  +  x2  —  impedance  of  the  circuit  of  (equivalent) 
resistance,  r,  and  (equivalent)  reactance,  x  =  2  Tr/L,  containing 
the  impressed  e.m.f.,  e0  and  the  counter  e.m.f.,  e\,  of  the  syn- 
chronous motor1;  that  is,  the  e.m.f.  generated  in  the  motor  arma- 
ture by  its  rotation  through  the  (resultant)  magnetic  field. 

Let  i  =  current  in  the  circuit  (effective  values). 

The  mechanical  power  delivered  by  the  synchronous  motor 
(including  friction  and  core  loss)  is  the  electric  power  consumed 
by  the  counter  e.m.f.,  e\]  hence 

p  =  id  cos  (i,  ei) ;  (1) 

thus, 


cos  (i,  61)  =  -r-j 


sin 


(2) 


The  displacement  of  phase  between  current  i,  and  e.m.f.  e  =  zi 
consumed  by  the  impedance,  z,  is 


cos  (i,  e)  =  - 
sin  (i,  e)  —  - 


(3) 


Since  the  three  e.m.fs.  acting  in  the  closed  circuit, 


eo  =  e.m.f.  of  generator, 

e\  —  counter  e.m.f.  of  synchronous  motor, 

e    =  zi  =  e.m.f.  consumed  by  impedance, 

1  If  eo  —  e.m.f.  at  motor  terminals,  z  =  internal  impedance  of  the  motor; 
if  eQ  =  terminal  voltage  of  the  generator,  z  =  total  impedance  of  line  and 
motor;  if  e0  =  e.m.f.  of  generator,  that  is,  e.m.f.  generated  in  generator 
armature  by  its  rotation  through  the  magnetic  field,  z  includes  the  generator 
impedance  also. 


316 


ALTERNATING-CURRENT  PHENOMENA 


form  a  triangle,  that  is,  e\  and  e  are  components  of  e0,  it  is  (Figs. 
159  and  160), 

eo2  =  6i2  +  e2  +  2  ee\  cos  (e\t  e),  (4) 


hence,  cos  (e\,  e)  = 


e0 


(5) 


since,  however,  by  diagram, 

cos  (ei,  e)  =  cos  (i,  e  —  i,  ei) 

=  cos  (i,  e)  cos  (i,  ei)  +  sin  (i,  e)  sin  (?',  61)       (6) 

substitution  of  (2),  (3)  and  (5)  in  (6)  gives,  after  some  trans- 
position, 

?i2  -  P2,  (7) 


e< 


2rp  =  2  x 


the  fundamental  equation  of  the  synchronous  motor,  relating  im- 
pressed e.m.f.,  e0;  counter  e.m.f.,  e\\  current,  i\  power,  p,  and  re- 
sistance, r;  reactance,  x;  impedance,  z. 


FIG.  159. 


FIG.  160. 


This  equation  shows  that,  at  given  impressed  e.m.f.,  eQ,  and 
given,  impedance,  z  =  VV2  +  x2,  three  variables  are  left,  e\,  i,  p, 
of  which  two  are  independent.  Hence,  at  given  e0  and  z,  the 
current,  i,  is  not  determined  by  the  load,  p,  only,  but  also  by  the 
excitation,  and  thus  the  same  current,  i,  can  represent  widely 
different  loads,  p}  according  to  the  excitation;  and  with  the  same 
load,  the  current,  i,  can  be  varied  in  a  wide  range,  by  varying  the 
field-excitation,  e\. 

The  meaning  of  equation  (7)  is  made  more  perspicuous  by 
some  transformations,  which  separate  e\  and  i,  as  function  of  p 
and  of  an  angular  parameter,  0. 

Substituting  in  (7)  the  new  coordinates; 

6!2  +  22i2 


a  = 


or, 


V 

r-/5 

VT 


a  — 


(8) 


SYNCHRONOUS  MOTOR 


we  get 


,2_ 


substituting  again,          e0"  =  a 
2zp  =  b 


hence, 

we  get 

a  — 
and,  squared, 

substituting 


r  = 

x  =  z\/l 
2  rp  =  e&, 


—    2 


317 

(9) 

(10) 


-  eb  =  V(l  -e2)(2a2-202-&2);  (11) 


M    ,    ^  -  e2)    ,    (a  -  €&)2 
a  -  eb)  -\ ^ j ~ 


(a  -e6)V2 
__^    _ 


gives,  after  some  transposition, 


a(a-2e6), 


hence,  if 


it  is 


(1  -  €2)  (a  -  2  e6)a 


=  0; 

(12) 

(13) 

(14) 

(15) 
(16) 


v2  +  w2  =  R2 

the  equation  of  a  circle  with  radius,  R. 

Substituting  now  backward,  we  get,  with  some  transpositions, 

{r2(ei2  +  z2*2)  -  z2(e02  -  2rp)J2  +  {rx(d2  +  Z2i2)}2  = 
Z2z2e02(eo2  -  4  rp)  (17) 

the  fundamental  equation  of  the  synchronous  motor  in  a  modified 
form. 

The  separation  of  e\  and  i  can  be  effected  by  the  introduction 
of  a  parameter,  0,  by  the  equations 

r2(ej2  +  zH2)  -  z2(eQ2  -  2  rp)  =  xzeQ\/eQ2  -  4  rp  cos  </>  I 


4  rp  sin 


These  equations  (18),  transposed,  give 


€l  =  \2  i  r2^2  ~ 


cos 


sn 


-  4  rp 


318         ALTERNATING-CURRENT  PHENOMENA 


r  . 


=\2  1  (r5  (e°2  " 


cos      ~  sn 


The  parameter,  0,  has  no  direct  physical  meaning,  apparently. 

These  equations  (19)  and  (20),  by  giving  the  values  of  e\  and  i 
as  functions  of  p  and  the  parameter,  <£,  enable  us  to  construct 
the  power  characteristics  of  the  synchronous  motor,  as  the  curves 
relating  e\  and  i,  for  a  given  power,  p}  by  attributing  to  $  all 
different  values. 

Since  the  variables,  v  and  w,  in  the  equation  of  the  circle  (16) 
are  quadratic  functions  of  e\  and  i,  the  power  characteristics  of 
the  synchronous  motor  are  quartic  curves. 

They  represent  the  action  of  the  synchronous  motor  under  all 
conditions  of  load  and  excitation,  as  an  element  of  power  trans- 
mission even  including  the  line,  etc. 

Before  discussing  further  these  power  characteristics,  some 
special  conditions  may  be  considered. 

220.  A.     Maximum  Output. 

Since  the  expression  of  d  and  i  [equations  (19)  and  (20)]  con- 
tain the  square  root,  \/eo2  —  4  rp,  it  is  obvious  that  the  maximum 
value  of  p  corresponds  to  the  moment  where  this  square  root 
disappears  by  passing  from  real  to  imaginary;  that  is, 

e02  —  4  rp  =  0, 
•  -  p  =  fr.      '  -     '••  (21) 

This  is  the  same  value  which  represents  the  maximum  power 
transmissible  by  e.m.f.,  e0,  over  a  non-inductive  line  of  resistance, 
r;  or,  more  generally,  the  maximum  power  which  can  be  trans- 
mitted over  a  line  of  impedance, 

z  =  Vr2  +  x2, 
into  any  circuit,  shunted  by  a  condenser  of  suitable  capacity. 

Substituting  (21)  in  (19)  and  (20),  we  get, 


(22) 


SYNCHRONOUS  MOTOR  319 

and  the  displacement  of  phase  in  the  synchronous  motor, 

/       -\        P       r 
cos  (ei,  i)  =  -r—  —  -; 
i&i      z 

hence, 

tan  (ely  i)  =  -  *,  (23) 

that  is,  the  angle  of  internal  displacement  in  the  synchronous 
motor  is  equal,  but  opposite  to,  the  angle  of  displacement  of  line 
impedance, 

(ei,  i)  =  -  (e,  i), 

-  (z,  r),  (24) 

and  consequently, 

(CQ,  i)  =  0;  (25) 

that  is,  the  current,  ?',  is  in  phase  with  the  impressed  e.m.f.,  e0. 

If  z  <  2  r,  ei  <  eoj  that  is,  motor  e.m.f.  <  generator  e.m.f. 
If  z  =  2  r,  ei  =  6o5  that  is,  motor  e.m.f.  =  generator  e.m.f. 
If  z  >  2  r,  e\  >  eQ]  that  is,  motor  e.m.f.  >  generator  e.m.f. 

In  either  case,  the  current  in  the  synchronous  motor  is  leading. 

221.  B.     Running  Light,  p  =  0. 

When  running  light,  or  for  p  =  0,  we  get,  by  substituting  in 
(19)  and  (20), 


g  COS  ^  +  z  sin  ^  f 


2  C°S      "" 


(26) 


Obviously  this  condition  cannot  well  be  fulfilled,  since  p  must 
at  least  equal  the  power  consumed  by  friction,  etc. ;  and  thus  the 
true  no-load  curve  merely  approaches  the  curve  p  =  0,  being, 
however,  rounded  off,  where  curve  (26)  gives  sharp  corners. 

Substituting  p  =  0  into  equation  (7)  gives,  after  squaring  and 
transposing, 

ei4+eo4-r-z4;4-2  eiW-2  zH2eQ*+2  z*i*ei*-4  xH2eiz  =  Q.  (27) 

This  quartic  equation  can  be  resolved  into  the  product  of  two 
quadratic  equations, 

ei2  +  z*i*  -  e02  +  2  xiel  =  0. 1  , 

ei2  +  z2*'2  -  e02  -  2  xiei  =  0. 


320         ALTERNATING-CURRENT  PHENOMENA 


which  are  the  equations  of  two  ellipses,  the  one  the  image  of  the 
other,  both  inclined  with  their  axes. 

The  minimum  value  of  counter  e.m.f  .,  ei,  is  e\  =  0  at  i  —  -  (29) 

The  minimum  value  of  current,  i,  is  i  =  0  at  e\  —  e0.          (30) 
The  maximum  value  of  e.m.f.,  ci,  is  given  from  equation    (28) 

/  =  6i2  -f  z2i2  -  eQz  ±  2  xie*  =  0; 


200- 


160- 


i 


\ 


gOOO        Mto 


Vofte  1000 


4000          6000 


\ 


\ 


B' 


V 


\ 


FIG.  161, 


by  the  condition, 

Ti* 

hence, 


df/di 
d}/del 


0,  as  zH  +  xei  =  0, 


X 

-, 


(31) 


SYNCHRONOUS  MOTOR  321 

The  maximum  value  of  current,  i,  is  given  from  equation  (28)  by 

as 

i  =  ~  ei  =  +  60  -.  (32) 

r  r 

If,  as  abscissas,  Ci,  and  as  ordinates,  zi,  are  chosen,  the  axes 
of  these  ellipses  pass  through  the  points  of  maximum  power 
given  by  equation  (22). 

It  is  obvious  thus,  that  in  the  V-shaped  curves  of  synchronous 
motors  running  light,  the  two  sides  of  the  curves  are  not  straight 
lines,  as  sometimes  assumed,  but  arcs  of  ellipses,  the  one  of  con- 
cave, the  other  of  convex,  curvature. 

These  two  ellipses  are  shown  in  Fig.  161,  and  divide  the  whole 
space  into  six  parts — the  two  parts,  A  and  A',  whose  areas  con- 
tain the  quartic  curves  (19)  (20)  of  the  synchronous  motor,  the 
two  parts,  B  and  #',  whose  areas  contain  the  quartic  curves  of  the 
generator,  the  interior  space,  C,  and  exterior  space,  D,  whose 
points  do  not  represent  any  actual  condition  of  the  alternator 
circuit,  but  make  e\  and  i  imaginary.  Some  of  the  quartic 
curves,  however,  may  overlap  into  space,  C. 

A  and  A'  and  the  same  B  and  Be  are  identical  conditions  of 
the  alternator  circuit,  differing  merely  by  a  simultaneous  reversal 
of  current  and  e.m.f.,  that  is  differing  by  the  time  of  a  half-period. 

Each  of  the  spaces  A  and  B  contains  one  point  of  equation  (22), 
representing  the  condition  of  maximum  output  as  generator,  viz., 
synchronous  motor. 

222.  C.     Minimum  Current  at  Given  Power. 

The  condition  of  minimum  current,  i,  at  given  power,  p,  is 
determined  by  the  absence  of  a  phase  displacement  at  the  im- 
pressed e.m.f.,  CD, 

(CD,  i)  =  0. 

This  gives  from  diagram  Fig.  160, 

ei2  =  eQ2  +  i2z2  -  2  ie0r,  (33) 

or,  transposed, 

ei    =  V(e0  -  irY  +  iV.  (34) 

This  quadratic  curve  passes  through  the  point  of  zero  current 
and  zero  power, 

i  =  0,      e\  —  Co, 
21 


322         ALTERNATING-CURRENT  PHENOMENA 

through  the  point  of  maximum  power  (22), 
.  _    0o  eQz 

=  2?    6l  ="   27 

and  through  the  point  of  maximum  current  and  zero  power, 

60  e0x 

*  =  ~,     ei  =  —>  (35) 

and  divides  each  of  the  quartic  curves  or  power  characteristics 
into  two  sections,  one  with  leading,  the  other  with  lagging,  cur- 
rent, which  sections  are  separated  by  the  two  points  of  equation 
(34),  the  one  corresponding  to  minimum,  the  other  to  maximum, 
current. 

It  is  interesting  to  note  that  at  the  latter  point  the  current 
can  be  many  times  larger  than  the  current  which  would  pass 
through  the  motor  while  at  rest,  which  latter  current  is, 


while  at  no-load  the  current  can  reach  the  maximum  value, 

'        '         '  '  •'' 


(36) 


(35) 


the  same  value  as  would  exist  in  a  non-inductive  circuit  of  the 
same  resistance. 

The  minimum  value  at  counter  e.m.f.,  e\,  at  which  coincidence 
of  phase,  (e0,  i)  =  0,  can  still  be  reached  is  determined  from  equa- 
tion (34)  by, 

T-* 

di 
as 

i  =  e0-2'  ei  =  e0--  (37) 

z2  z 

The  curve  of  no-displacement,  or  gf  minimum  current,  is  shown 
in  Figs.  161  and  162  in  dotted  lines.1 

1  It  is  interesting  to  note  that  the  equation  (34)  is  similar  to  the  value 
ei  =  V(e0  —  ir)2  —  i2x2,  which  represents  the  output  transmitted  over  an 
inductive  line  of  impedance,  z  =  vV2  +  z2,  into  a  non-inductive  circuit. 

Equation  (34)  is  identical  with  the  equation  giving  the  maximum  voltage, 
«i,  at  current,  i,  which  can  be  produced  by  shunting  the  receiving  circuit  with 
a  condenser;  that  is,  the  condition  of  "complete  resonance"  of  the  line,  z  — 

vr*  +  x2,  with  current,  t.     Hence,  referring  to  equation  (35),  e\  =  e<r  is 

the  maximum  resonance  voltage  of  the  line  reached  when  closed  by  a  con- 
denser of  reactance,  —  z. 


SYNCHRONOUS  MOTOR  323 

223.  D.     Maximum  Displacement  of  Phase. 

(0o>  fc)  =  maximum. 
At  a  given  power,  p,  the  input  is, 

PQ  =  p  +  izr  =  e0i  cos  (eQ,  i)  ;  (38) 

hence'  cos  K  »)  -  £±4^-  (39) 

d(fli 

At  a  given  power,  p,  this  value,  as  function  of  the  current,  i, 
is  a  maximum  when 


e0i 
this  gives, 

p  =  i*r,  (40) 

or'  • 


That  is,  the  displacement  of  phase,  lead  or  lag  is  a  maximum 
when  the  power  of  the  motor  equals  the  power  consumed  by  the 
resistance;  that  is,  at  the  electrical  efficiency  of  50  per  cent. 

Substituting  (40)  in  equation  (7)  gives,  after  squaring  and 
transposing,  the  quartic  equation  of  maximum  displacement, 

(eo2  -  ei2)2  +  *4z2(z2  +  8  r2)  +  2  taeia(4  r2  -  z2)  - 

2  i'aeo  V  +  3  r2)  =  0.     (42) 

The  curve  of  maximum  displacement  is  shown  in  dash-dotted 
lines  in  Figs.  161  and  162.  It  passes  through  the  point  of  zero 
current  —  as  singular  or  nodal  point  —  and  through  the  point  of 
maximum  power,  where  the  maximum  displacement  is  zero,  and 
it  intersects  the  curve  of  zero  displacement. 

224.  E.     Constant  Counter  e.m.f. 

At  constant  counter  e.m.f.,  e\  =  constant. 

If  e0x 

&i  <~  eo  —  —  - 

the  current  at  no-load  is  not  a  minimum,  and  is  lagging.  With 
increasing  load  the  lag  decreases,  reaches  a  minimum,  and  then 
increases  again,  until  the  motor  falls  out  of  step,  without  ever 
coming  into  coincidence  of  phase. 

"  ; 


324         ALTERNATING-CURRENT  PHENOMENA 

the  current  is  lagging  at  no-load.  With  increasing  load  the  lag 
decreases,  the  current  comes  into  coincidence  of  phase  with  e0, 
then  becomes  leading,  reaches  a  maximum  lead;  then  the  lead 
decreases  again,  the  current  comes  again  into  coincidence  of 
phase,  and  becomes  lagging,  until  the  motor  falls  out  of  step. 

If  eQ  <  ei,  the  current  is  leading  at  no-load,  and  the  lead  first 
increases,  reaches  a  maximum,  then  decreases;  and  whether  the 
current  ever  comes  into  coincidence  of  phase  and  then  becomes 
lagging,  or  whether  the  motor  falls  out  of  step  while  the  current 
is  still  leading,  depends  whether  the  counter  e.m.f.  at  the  point 
of  maximum  output  is  >  eQ  or  <  e<>. 

225.  F.     Numerical  Example. 

Figs.  161  and  162  show  the  characteristics  of  a  100-kw.  motor 
supplied  from  a  2500- volt  generator  over  a  distance  of  5  miles, 
the  line  consisting  of  two  wires,  No.  2  B.  &  S.,  18  in.  apart. 

In  this  case  we  have: 


CQ  =  2500  volts  constant  at  generator  terminals; 
r  =      10  ohms,  including  line  and  motor; 
x  =      20  ohms,  including  line  and  motor; 
hence  z  =  22.36  ohms. 

Substituting  these  values,  we  get: 


(43) 


25002   -  eS  -  500  i2  -  20  p  =  40  VV  -  p2  (7) 

ei2  +  500  i2  -  31.25  X  106  +  100  p)2  -f  {2  ef  -  1000  i2)2  = 

7.8125  X  1014  -  5  X  109  p.  (17) 

i  =  5590  X  (19) 

3.2  X  10-6  p)  +  (0.894  cos  0  +  0.447  sin  0) 


Vl  -  6.4  X  10-6  p  \ .        (20) 
i  =  250  X 


(1  -  3.2  X  10~6p)  +  (0.894  cos  0-0.447  sin  0)Vl6.4XlO-6p). 

Maximum  output, 

p  =  156.25  kw.  (21) 

at  ei  =  2795  volts  .^ 

i  =  125  amp. 
Running  light, 

ei2  +  500  iz  -  6.25  X  104  +  40  iei  =  0  1  ,2g, 

ei  =  20  i  ±  V6.25  X  104  -  100  z2 


SYNCHRONOUS  MOTOR 


325 


At  the  minimum  value  of  counter  e.m.f.,  e\  —     0  is  i  — 112  (29) 

At  the  minimum  value  of  current,  i  —     0  is  e\  —  2500  (30) 

At  the  maximum  value  of  counter  e.m.f.,  e\  —  5590  is  i  =  223.5  (31) 

At  the  maximum  value  of  current,  i  =  250  is  e\  —  5000.  (32) 

Curve  of  zero  displacement  of  phase, 

61  =  10  V(250  -  i}2  +  4i2  (34) 


=  10  V6.25  X  104  -  500  i  +  5  i2. 


260 


180 


ISO 


120 


>7 


Vo'l   3 


600    1000    150U    2000    2500   '2000    3500    1000    41500    6000    6500 

FIG.  162. 

Minimum  counter  e.m.f.  point  of  this  curve, 

i  =  50,         0!  =  2240.  (35) 

Curve  of  maximum  displacement  of  phase, 

p  =  10  i2  (40) 

(6.25  X  106  -  d2)2  +  0.65  X  106  i4  -  1010  i2  =  0       (42) 


326 


ALTERNATING-CURRENT  PHENOMENA 


Fig.  161  gives  the  two  ellipses  of  zero  power  in  full  lines, 
with  the  curves  of  zero  displacement  in  dotted,  the  curves  of 
maximum  displacement  in  dash-dotted  lines,  and  the  points  of 
maximum  power  as  crosses. 

Fig.  162  gives  the  motor-power  characteristics  for  p  =  10  kw.; 
p  =  50  kw.;  p  =  100  kw.;  p  =  150  kw.,  and  p  =  156.25  kw., 
together  with  the  curves  of  zero  displacement  and  of  maximum 
displacement. 

226.  G.     Discussion  of  Results. 

The  characteristic  curves  of  the  synchronous  motor,  as  shown 
in  Fig.  162,  have  been  observed  frequently,  with  their  essential 
features,  the  V-shaped  curve  of  no-load,  with  the  point  rounded 
off  and  the  two  legs  slightly  curved,  the  one  concave,  the  other 


140 


120 


L 


100 


80 


1.60 
40 


20 


\ 


500 


1000 


1500 


D    2500 
Volts 

FIG.  163. 


3000    3500 


4000 


4500 


5000 


convex;  the  increased  rounding  off  and  contraction  of  the  curves 
with  increasing  load;  and  the  gradual  shifting  of  the  point  of 
minimum  current  with  increasing  load,  first  toward  lower,  then 
toward  higher,  values  of  counter  e.m.f.,  e\. 

The  upper  parts  of  the  curves,  however,  I  have  never  been 
able  to  observe  completely  and  consider  it  as  probable  that 
they  correspond  to  a  condition  of  synchronous  motor  running, 
which  is  unstable.  The  experimental  observations  usually 


SYNCHRONOUS  MOTOR  327 

extend  about  over  that  part  of  the  curves  of  Fig.  162  which  is 
reproduced  in  Fig.  163,  and  in  trying  to  extend  the  curves 
further  to  either  side,  the  motor  is  thrown  out  of  synchronism. 

It  must  be  understood,  however,  that  these  power  charac- 
teristics of  the  synchronous  motor  in  Fig.  162  can  be  considered 
as  approximations  only,  since  a  number  of  assumptions  are  made 
which  are  not,  or  only  partly,  fulfilled  in  practice.  The  fore- 
most of  these  are: 

1.  It  is  assumed  that  e\  can  be  varied  unrestrictedly,  while 
in  reality  the  possible  increase  of  e\  is  limited  by  magnetic 
saturation.     Thus   in   Fig.  162,  at   an  impressed   e.m.f.,   eQ  = 
2500  volts,  d  rises  up  to  5590  volts,  which  may  or  may  not  be 
beyond  that  which  can  be  produced  by  the  motor,  but  certainly 
is  beyond  that  which  can  be  constantly  given  by  the  motor. 

2.  The   reactance,    x,   is    assumed   as   constant.     While   the 
reactance  of  the  line  is  practically  constant,  that  of  the  motor 
is  not,  but  varies  more  or  less  with  the  saturation,  decreasing 
for  higher  values.     This  decrease  of  x  increases  the  current,  i, 
corresponding  to  higher  values  of  e\t  and  thereby  bends  the  curves 
upward  at  a  lower  value  of  e\  than  represented  in  Fig.  162. 

It  must  be  understood  that  the  motor  reactance  is  not  a 
simple  quantity,  but  represents  the  combined  effect  of  self- 
induction,  that  is,  the  e.m.f.  generated  in  the  armature  con- 
ductor by  the  current  therein  and  armature  reaction,  or  the 
variation  of  the  counter  e.m.f.  of  the  motor  by  the  change  of 
the  resultant  field,  due  to  the  superposition  of  the  m.m.f.  of 
the  armature  current  upon  the  field-excitation ;  that  isr  it  is  the 
"synchronous  reactance." 

3.  Furthermore,  this  synchronous  reactance  usually  is  not  a 
constant  quantity  even  at  constant  induced  e.m.f.,  but  varies 
with  the  position  of  the  armature  with  regard  to  the  field;  that 
is,  varies  with  the  current  and  its  phase  angle,  as  discussed  in  the 
chapter  on  the  armature  reactions  of  alternators.     While  in 
most  cases  the  synchronous  reactance  can  be  assumed  as  con- 
stant, with  sufficient  approximation,  sometimes  a  more  com- 
plete investigation  is  necessary,  consisting  in  a  resolution  of  the 
synchronous  impedance  in  two  components,  in  phase  and  in 
quadrature  respectively  with  the  field-poles. 

Especially  is  this  the  case  at  low  power-factors.  So  by 
gradually  decreasing  the  excitation  and  thereby  the  e.m.f.,  e, 
the  curves  may,  especially  at  light  load,  occasionally  be  extended 


328         ALTERNATING-CURRENT  PHENOMENA 

below  zero,  into  negative  values  of  e,  or  onto  the  part  of  the 
curve,  By  in  Fig.  161,  while  the  power  still  remains  constant 
and  positive,  as  synchronous  motor.  In  other  words,  the  motor 
keeps  in  step  even  if  the  field-excitation  is  reversed;  the  lagging 
component  of  the  armature  reaction  magnetizes  the  field,  in 
opposition  to  the  demagnetizing  action  of  the  reversed  field 
excitation. 

4.  These  curves  in  Fig.  162  represent  the  conditions  of  con- 
stant electric  power  of  the  motor,  thus  including  the  mechan- 
ical and  the  magnetic  friction  (core  loss).  While  the  mechanical 
friction  can  be  considered  as  approximately  constant,  the  mag- 
netic friction  is  not,  but  increases  with  the  magnetic  induction; 
that  is,  with  e\,  and  the  same  holds  for  the  power  consumed  for 
field  excitation. 

Hence  the  useful  mechanical  output  of  the  motor  will  on  the 
same  curve,  p  =  const.,  be  larger  at  points  of  lower  counter 
e.m.f.,  eit  than  at  points  of  higher  e\\  and  if  the  curves  are 
plotted  for  constant  useful  mechanical  output,  the  whole  system 
of  curves  will  be  shifted  somewhat  toward  lower  values  of  e\\ 
hence  the  points  of  maximum  output  of  the  motor  correspond 
to  a  lower  e.m.f.  also. 

It  is  obvious  that  the  true  mechanical  power  characteristics 
of  the  synchronous  motor  can  be  determined  only  in  the  case  of 
the  particular  conditions  of  the  installation  under  consideration. 

227.  H.  Phase  Characteristics  of  the  Synchronous  Motor. 
I  While  an  induction  motor  at  constant  impressed  voltage  is 
fully  determined  as  regards  to  current,  power-factor,  efficiency, 
etc.,  by  one  independent  variable,  the  load  or  output;  in  the 
synchronous  motor  two  independent  variables  exist,  load  and 
field-excitation.  That  is,  at  constant  impressed  voltage  the 
current,  power-factor,  etc.,  of  a  synchronous  motor  can  at  the 
same  power  output  be  varied  over  a  wide  range  by  varying 
the  field-excitation,  that  is,  the  counter  e.m.f.  or  "nominal  gener- 
ated e.m.f."  Hence  the  synchronous  motor  can  be  utilized  to 
fulfill  two  independent  functions:  to  carry  a  certain  load  and  to 
produce  a  certain  wattless  current,  lagging  by  under-excitation, 
leading  by  over-excitation.  Synchronous  motors  are,  therefore, 
to  a  considerable  extent  used  to  control  the  phase  relation  and 
thereby  the  voltage,  in  addition  to  producing  mechanical  power. 

The  same  applies  to  synchronous  converters. 

With  given  impressed  e.m.f.,  field-excitation  or  nominal  gener- 


SYNCHRONOUS  MOTOR 


329 


ated  e.m.f.  corresponding  thereto,  and  load,  determine  all  the 
quantities  of  the  synchronous  motor,  as  current,  power-factor, 
etc.  Thus  if  in  diagram  Fig.  164,  OE  =  e  =  e.m.f.  consumed  by 
the  counter  e.m.f.  or  nominal  generated  e.m.f.  of  the  synchronous 
motor,  and  if  PQ  =  output  of  motor  (exclusive  of  friction  and  core 
loss  and,  if  the  exciter  is  driven  by  the  motor,  power  consumed 

r> 

by  the  exciter),  i\  —  —  =  power  component  of  current,  repre- 

v 

sented  by  01 1,  and  the  current  vector  therefore  must  terminate 
on  a  line,  i,  perpendicular  to  01 1.  If,  then,  r  =  resistance  and 
x  =  reactance  of  the  circuit  between  counter  e.m.f.,  e,  and  im- 


FIG.  164. 

pressed  e.m.f.,  CQ,  OEr  =  i-p  =  e.m.f.  consumed  by  resistance, 
OEX  =  iix  =  e.m.f.  consumed  by  reactance  of  the  power  com- 
ponent of  the  current,  i\,  hence  OE'i  =  e.m.f.  consumed  by 
impedance  of  the  power  component  of  the  current,  i\,  and  the 
impedance  voltage  of  the  total  current  lies  on  the  perpendicular 
e'  on  OE'i.  Producing  OEi  =  OE}  and  drawing  an  arc  with 
the  impressed  e.m.f.,  e0,  as  radius  and  E\  as  center,  the  point 
of  intersection  with  e'  gives  the  impedance  voltage,  OE',  and 
corresponding  thereto  the  current  01  =  i\  and  completing  the 
parallelogram,  OEEQE',  gives  the  impressed  e.m.f.,  OEQ. 

Hence,  by  impressed  e.m.f.,  e0,  counter  e.m.f.,  e,  and  load,  Po, 
the  vector  diagram  is  determined,  and  thereby  the  vectors,  01  = 


330         ALTERNATING-CURRENT  PHENOMENA 

current,    OE0  =  impressed    e.m.f.,    OE  =  counter    e.m.f.,    and 
their  phase  relation. 

Or,  in  symbolic  representation,  let 

EQ  =  e\  —  je"Q  =  impressed  e.m.f.; 


eQ  =  VV2  +  eo"2;  (1) 

E  =  e'  —  je"  =  e.m.f.  consumed  by  counter  e.m.f.; 

e  =  Ve/2-M"2;  (2) 

l=i  =  current,  assumed  as  zero  vector; 
Z  =  r  +  jx  =  impedance  of  circuit  between  eQ  and  e. 
Z  is  the  synchronous  impedance  of  the  motor,  if  eQ  is  its  ter- 
minal voltage.     It  is  the  impedance  of  transmission  line  with 
transformers  and  motor,  if  eQ  is  terminal  voltage  of  generator,  and 
Z  is  synchronous  impedance  of  motor  and  generator,  plus  impe- 
dance of  line  and  transformers,  if  eo  is  the  nominal  generated 
e.m.f.  of  the  generator  (corresponding  to  its  field-excitation). 
It  is,  then, 

EQ  =  E  +  %Zt  (3) 

or, 

e'o  -  je"Q  =  er  -  je"  +  ir  +  jix,  (4) 

and,  resolved, 

Vo    =  e'   +  ir;  (5) 

e"0  =  e"  -  ix.  (6) 

The  power  output  of  the  motor  (inclusive  of  friction  and  core 
loss,  and  if  the  exciter  is  driven  by  the  motor,  power  consumed 
by  exciter)  is  current  times  power  component  of  generated 
e.m.f.,  or 

Po  =  e'i.  (7) 

Hence,  the  calculation  of  the  motor,  of  supply  voltage  eQ 
from  power  output,  PQj  occurs  by  the  equations: 

Chosen:  i  =  current. 


(7)  e'     =      , 

(5)  e'0    =  e'  +  ir, 


(1)  e",  =  ± 

(6)  e"    =  e",  +  ix 

(2)  e      =  V^M1  € 


(8) 


That  is,  at  given  power,  P0,  to  every  value  of  current,  i,  corre- 
spond two  values  of  the  counter  e.m.f.,  e  (and  hence  the  field- 
excitation). 


SYNCHRONOUS  MOTOR 


331 


Solving  equations   (8)  for  i  and  P0,  that  is,  eliminating  e', 
e'o,  e"0,  e",  gives  as  the  nominal  generated  e.m.f., 

/     9  9-91         9-9          r»r>if>-/9  /*•      i          A  2      (Q\ 

e  =  A/CO    —  fHa  +  JC*la  —  2  rr0  +  2  xi  A/^O    —  l~ — r  r»l  ,  vy/ 

and  the  power-factor  of  the  motor  is, 

f>.r       P 

(10) 


COS 


ei 


The  power-factor  of  the  supply  is 


cos  0o  = 


Po    ,    . 
e^o  =    ^   +tr  =  Po  +  r^2 

#o  60  60^' 


(ID 


From  equation  (9),  by  solving  for  i,  i  can  now  be  expressed  as 
function  of  P0  and  e,  that  is,  of  power  output  and  field-excitation. 


200  400  COO  800  1000  1200 1400 1600  1800  2000  2200  2400  2COO-2800  8000  8200  3400  3COO  880040004200 

VOLTS  =  6 

FIG.  165. 

248.  As  illustrations  are  plotted,  in  Fig.  165,  curves  giving  the 
current,  i,  as  function  of  the  counter  or  nominal  generated  e.m.f., 
e,  at  constant  power,  P0.  Such  curves  as  discussed  before  in  Figs. 
161,  162,  163,  are  called  "phase  characteristics  of  the  synchro- 

nniic  Tr»/-kfr»T«  *' 


nous  motor." 


332         ALTERNATING-CURRENT  PHENOMENA 

They  are  given  for  the  values 

60  =  2200  volts, 
Z  =  1  +  4  j  ohms, 
and 

Po  =  20,  200,  400,  600,  800,  1000  kw.  output. 

The  five  equations  of  the  synchronous  motor, 

(1)  e02    =  e0'2  +  e0"2, 

(2)  e2    =  e'2  +  e"2, 
(7)  Po    =  e'i, 

(5)  e'Q    =  e'  +  ir, 

(6)  e"0  =  e"  -  ix, 

determine  the  five  quantities,  e'0,  e"0,  e',  e" ',  e,  as  functions  of  P0 
and  i. 

The  condition  of  zero  phase  displacement,  or  unity  power- 
factor  at  the  impressed  e.m.f.,  e0,  is 

e".  =  0; 

hence  e'0  =  eQ, 

and  (6)  e"  =  is, 

(5)  e'  =  60  -  ir; 
hence, 

e2  =  (CQ  -  irY  +  «,  (12) 

a   quadratic   equation,    the   hyperbola   of   unity   power-factor, 
shown  as  dotted  line  in  Fig.  165. 

In  this  case,  the  power  is  found  by  substituting  e'  =  e0  —  ir 
in  Po  =  e'  i,  as 

Po  -  e0i  -  izr,  (13) 

or  

47P01  ( 


The  maximum  output  of  the  synchronous  motor  follows  here- 
from,  by  the  condition, 


in  above  example 

Pm  =  1210  kw.  at  i  =  1100  amp. 


SYNCHRONOUS  MOTOR 


333 


| 

•^v 

-X 

^ 

fA 

cyp 

^ 

X 

/ 

" 

^t 

/Vcy 

s 

z 

/ 
t 

^ 

\ 

\ 

•/ 

I 

^J 

.^-  —  ' 

.—  —  — 

—  -^ 

^v 

s\ 

1 

// 

/ 

Nj 

\ 

sA 

1 

* 

7 

/ 

N. 

S  500 

1 

/ 

£ 

/ 

/ 

1 

A 

of 

/ 

/ 

400 

1    1 
I 

HI 

/ 

/ 

/ 

// 

/ 

800 

1 

A 

1 

X 

x 

V 

/ 

-  f> 

^ 

>1 

PO 

=2. 

00 

V 

/, 

/ 

e  = 

=  re 

oo 

V 

fl 

y 

.  —  • 

—  • 

^^ 

z 

=H 

;4j 

'7 

P' 

-2 

3KV 

7 

0 

0 

100          200         800          400         600          600         700 
KILOWATTS 

FIG.  166. 


FIG.  167. 


334         ALTERNATING-CURRENT  PHENOMENA 

The  curve  of  unity  power-factor  (12)  divides  the  synchronous 
motor-phase  characteristics  into  two  sections,  one,  for  lower  e, 
with  lagging,  the  other  with  leading  current. 

The  study  of  these  "phase  characteristics,'7  Fig.  165,  gives  the 
best  insight  into  the  behavior  of  the  synchronous  motor  under 
conditions  of  steady  operation. 


400          500  600 

KILOWATTS 


900 


FIG.  168. 


229.  I.  Load  Curves  of  Synchronous  Motor. 

Of  special  interest  are  the  "load  curves"  of  the  synchronous 
motor,  or  curves  giving,  at  constant  excitation,  e  —  constant, 
the  current,  power-factor,  efficiency  and  apparent  efficiency  as 


SYNCHRONOUS  MOTOR 


335 


function  of  the  load  or  output  P  —  PQ  —  (friction  +  core  loss  + 
excitation).  Such  load  curves  are  represented  in  Figs.  166  to 
170,  for  e  =  1600,  2000,  2180,  2400,  2800  volts.  They  can 
be  derived  from  Fig.  165  as  the  intersection  of  the  curves  P0  = 
constant  with  the  vertical  lines  e  =  constant. 

Hence,  while  an  induction  motor  has  one  load  curve  only,  a 
synchronous  motor  has  an  infinite  series  of  load  curves,  depend- 
ing upon  the  value  of  e. 


1000 


1002 


400          500         600 
KILOWATTS 

FIG.  169. 


800        900 


For  low  values  of  e  (e  =  1600,  under  excitation,  Fig.  166), 
the  load  curves  are  similar  to  those  of  an  induction  motor. 
The  current  is  lagging,  the  power-factor  rises  from  a  low  initial 
value  to  a  maximum,  and  then  falls  again.  With  increasing 
excitation  (e  =  2000,  Fig.  167)  the  power-factor  curve  rises  to 
values  beyond  those  available  in  induction  motors,  approaches 
and  ultimately  touches  unity,  and  with  still  higher  excitation 
(e  =  2180,  Fig.  168)  two  points  of  unity  power-factor  exist,  at 
P  =  20  and  P  =  450  kw.  output,  which  are  separated  by  a 
range  with  leading  current,  while  at  very  low  and  very  high  load 
the  current  is  lagging.  The  first  point  of  unity  power-factor 


336 


ALTERNATING-CURRENT  PHENOMENA 


moves  toward  P  =  0,  and  then  disappears,  that  is,  the  current 
becomes  leading  already  at  no-load,  and  the  second  point  of 
unity  power-factor  moves  with  increasing  excitation  toward 
higher  loads,  from  P  =  450  kw.  at  e  =  2180  in  Fig.  168,  to  P  = 
700  kw.  at  e  =  2400,  Fig.  169,  and  P  =  900  kw.  at  e  =  2800, 
Fig.  170,  while  the  power-factor  and  thereby  the  apparent 
efficiency  decrease  at  light  loads.  The  maximum  output  in- 
creases with  the  increase  of  excitation  and  almost  proportionally 
thereto. 


000 
800 
700 

% 

100 

90 
30 
70 
fiO 
50 
40 
30 
20 
10 

^- 

—  " 

) 

^ 

/ 

,& 

y 

EPl 

irvr 

\ 

^ 

/•- 

-_ 

/' 

•  "* 

& 

y 

^ 

,  ' 

.  —  - 

^> 

^x 

I 

/ 

/ 

/ 

^ 

y 

/ 

/ 

/ 

i 

f 

\ 

H" 

L 

400 
800 
200 
100 
0 

1 

/ 

4 

/ 

/ 

j 

1 

1 

/ 

4 

/ 

/ 

/ 

t 
1 

/ 

IP/ 

/ 

/ 

<# 

^ 

/ 

1 

/ 

I 

.r>\ 

,*£ 

^ 

; 

/ 

/ 

^ 

£>• 

V 

/ 

„--• 

^ 

. 

I 

/ 

^ 

^ 

9 

PO- 

'2'. 

00 

V 

l\ 

u 

•*•" 

^"^ 

e  = 

'2800 

V 

z.  = 

•1+4J 

\l 

P^ 

*2£ 

KV\ 

I 

100         200 


400         600        600 
KILOWATTS 

FIG.  170. 


700        800         900       1000 


It  is  interesting  that  at  e  =  2180,  the  power-factor  is  practi- 
cally unity  over  the  whole  range  of  load  up  to  near  the  maximum 
output.  It  is  shown  once  more  in  Fig.  168  with  increased  scale 
of  the  ordinates.  A  synchronous  motor  at  constant  excitation 
can,  therefore,  give  practically  unity  power-factor  for  all  loads. 

The  resistance,  r  =  1  ohm,  is  assumed  so  as  to  represent  a  syn- 
chronous motor  inclusive  of  transmission  line,  with  about  9  per 
cent,  loss  of  energy  in  the  line  at  400  kw.  output. 

The  friction  and  core  loss  are  assumed  as  20  kw.,  the  excitation 
as  4  kw.  at  e  =  2000. 


SYNCHRONOUS  MOTOR 


337 


Considering  the  intersections  of  a  horizontal  line  with  the 
constant  power  curves  of  Fig.  165,  gives  the  characteristic  curves 
of  the  synchronous  motor  when  operating  on  constant  current. 
Such  curves  are  shown  for  i  =  300  in  Fig.  171.  They  illustrate 


8400 


2200 
2000 
1800 
1600 
1400 
1200 


800 


ONSTANT  CURR 


F'=2;o+|e 


NT  SYNCHRONOUS 


=220 


10 


OR 


\ 


200  300  400 

KILOWATTS 


100 


70 


FIG.  171. 

that  at  the  same  impressed  voltage,  with  the  same  current  input 
the  power  output  of  the  synchronous  motor  can  vary  over  a  wide 
range,  and  also  that  for  each  value  of  power  output  two  points 
exist,  one  with  lagging,  the  other  with  leading  current. 

22 


338         ALTERNATING-CURRENT  PHENOMENA 

As  regards  phase  characteristics  and  load  characteristics, 
the  same  applies  to  the  synchronous  converter  as  to  the  syn- 
chronous motor,  except  that  in  the  former  the  continuous  cur- 
rent output  affords  a  means  of  automatically  varying  the 
excitation  with  the  load. 

230.  The  investigation  of  a  variation  of  the  armature  reaction 
and  the  self-induction,  that  is,  of  the  synchronous  reactance, 
with  the  position  of  the  armature  in  the  magnetic  field,  and  so 
the  intensity  and  phase  of  the  current  in  its  effect  on  the  charac- 
teristic curves  of  the  synchronous  motor,  can  be  carried  out  in 
the  same  manner  as  done  for  the  alternating-current  generator 
in  Chapter  XX. 

In  the  graphical  and  the  symbolic  investigations  in  Chapter 
XX,  the  current,  /  =  i\  —  jiz,  has  been  considered  as  the 
output  current,  and  chosen  of  such  phase  as  to  differ  less  than 
90°  from  the  terminal  voltage,  E  =  e\  +  je^,  so  representing 
power  output. 

Choosing  then  the  current  vector,  01,  in  opposite  direction  from 
that  chosen  in  Figs.  139  and  140,  and  then  constructing  the 
diagram  in  the  same  manner  as  done  in  Chapter  XX,  brings  the 
output  current,  01,  more  than  90°  displaced  from  the  terminal 
voltage,  OE.  Then  the  current  consumes  power,  that  is,  the 
machine  is  a  synchronous  motor.  The  graphical  representation 
in  Chapter  XX  so  applies  equally  well  to  alternating-current 
generator  as  to  synchronous  motor,  and  the  former  corresponds 
to  the  case  Z  EOI  <  90°,  the  latter  to  the  case:  Z  EOI  >  90°. 

In  the  same  manner,  in  the  symbolic  representation  of  Chapter 
XX,  choosing  the  current  as  I  =  —  i\  +  jiz,  or,  in  the  final 
equation,  where  the  current  has  been  assumed  as  zero  vector, 
/  =  —  i,  that  is,  reversing  all  the  signs  of  the  current,  gives  the 
equations  of  the  synchronous  motor. 

Choosing  the  same  denotations  as  in  Chapter  XX,  and  sub- 
stituting —  i  for  +  i  in  equation  (64)  so  gives  the  general 
equation  of  the  synchronous  motor, 

(ei  —  n')2 

V(ei- 
and  for  non-inductive  load, 

=  (e  -r 

60  ~  V(e  - 


SYNCHRONOUS  MOTOR  339 

Or,  by  choosing  01  in  the  graphic,  and  I  =  I'  +  /"  in  the 
symbolic  method,  as  the  input  current,  the  diagram  can  be 
constructed  by  combining  the  vectors  in  their  proper  directions, 
that  is,  where  they  are  added  in  Chapter  XX,  they  are  now 
subtracted,  and  inversely.  For  instance, 

Ei  =  E2  +  Ei,        E  =  Ei  +  #4,  etc. 

The  reversal  of  the  sign  of  the  current  in  the  above  equations, 
compared  with  the  equations  of  Chapter  XX,  shows  that  in  the 
synchronous  motor,  the  effect  of  lag  and  of  lead  of  the  input 
current  are  the  opposite  of  the  effect  of  lag  and  lead  of  the  output 
current  in  the  generator,  as  discussed  before. 

It  also  follows  herefrom,  that  the  representation  of  the  internal 
reactions  of  the  synchronous  motor  by  an  effective  reactance, 
the  " synchronous  reactance,"  is  theoretically  justified;  but  that, 
like  in  the  alternating-current  generator,  this  reactance  may  have 
to  be  resolved  in  two  components,  x'Q  and  x",  parallel  and  at 
right  angles  respectively  to  the  field-poles. 

231.  The  phase  characteristics,  Fig.  165,  and  more  particularly 
the  no-load  curve,  is  of  special  importance  in  the  so-called  syn- 
chronous condenser,  that  is,  a  synchronous  machine  running  idle 
and  producing  lagging  or  leading  current  at  will. 

As  at  constant  impressed  voltage,  the  reactive  current  taken 
by  the  synchronous  machine  depends  upon,  and  varies  with  the 
field-excitation,  synchronous  motors  offer  a  convenient  means  for 
producing  reactive  currents  of  varying  amounts. 

As  lagging  reactive  currents  can  more  conveniently  be  pro- 
duced by  stationary  reactors,  synchronous  machines  are  mainly 
used  for  producing  leading  currents,  or  producing  reactive  cur- 
rents varying  between  lag  and  lead.  Therefore,  the  name 
"synchronous  condenser"  for  such  machines. 

Their  foremost  use  is : 

1.  For   power-factor   correction   in   systems   of   low   power- 
factor,  such  as  systems  containing  many  induction  motors  or 
other  reactive  devices.     In  this  case,  the  synchronous  condenser 
is  connected  in  shunt  to  the  circuit  as  close  to  the  source  of  the 
reactive  lagging  currents  as  feasible. 

2.  For  voltage   control   of  long-distance  transmission  lines. 
In  very  long  lines,  especially  at  60  cycles,  the  inherent  voltage 
regulation  at  the  receiving  end  of  the  line  becomes  very  poor, 
and  then  a  synchronous  condenser  is  made  to  "float"  on  the 


340         ALTERNATING-CURRENT  PHENOMENA 

receiving  circuit,  controlled  by  a  voltage  regulator  so  that  its 
reactive  current  varies  from  lag  at  no-load  on  the  line,  to  lead  at 
heavy  load,  and  thereby  maintains  the  line  voltage  constant. 

In  synchronous  condensers,  low  armature  reaction  is  an  ad- 
vantage, as  requiring  less  field  regulation. 

As  synchronous  condensers  must  run  at  high  leading  currents, 
and  this  is  the  condition  where  the  tendency  to  surging  is  greatest, 
synchronous  condensers  are  usually  supplied  with  anti-hunting 
devices.  For  this  purpose,  generally  a  squirrel-cage  winding  in 
the  field-poles  is  used.  Such  a  winding  is  desirable  also  to 
improve  the  self-starting  character  of  the  machine. 

Very  large  synchronous  condensers  are  in  successful  operation 
on  transmission  lines  of  such  length,  that  without  the  syn- 
chronous condenser,  operation  of  the  circuits  would  be  entirely 
impossible. 


SECTION  VI 
GENERAL  WAVES 


CHAPTER  XXV 
DISTORTION  OF  WAVE-SHAPE  AND  ITS  CAUSES 

232.  In  the  preceding  chapters  we  have  considered  the  alter- 
nating  currents  and   alternating  e.m.fs.   as  sine   waves  or  as 
replaced  by  their  equivalent  sine  waves. 

While  this  is  sufficiently  exact  in  most  cases,  under  certain 
circumstances  the  deviation  of  the  wave  from  sine  shape  becomes 
of  importance,  and  with  certain  distortions  it  may  not  be  pos- 
sible to  replace  the  distorted  wave  by  an  equivalent  sine  wave, 
since  the  angle  of  phase  displacement  of  the  equivalent  sine 
wave  becomes  indefinite.  Thus  it  becomes  desirable  to  investi- 
gate the  distortion  of  the  wave,  its  causes  and  its  effects. 

Since,  as  stated  before,  any  alternating  wave  can  be  repre- 
sented by  a  series  of  sine  functions  of  odd  orders,  the  inves- 
tigation of  distortion  of  wave-shape  resolves  itself  in  the  in- 
vestigation of  the  higher  harmonics  of  the  alternating  wave. 

In  general  we  have  to  distinguish  between  higher  harmonics 
of  e.m.f.  and  higher  harmonics  of  current.  Both  depend  upon 
each  other  in  so  far  as  with  a  sine  wave  of  impressed  e.m.f.  a 
distorting  effect  will  cause  distortion  of  the  current  wave,  while 
with  a  sine  wave  of  current  passing  through  the  circuit,  a  dis- 
torting effect  will  cause  higher  harmonics  of  e.m.f. 

233.  In  a  conductor  revolving  with  uniform  velocity  through 
a  uniform  and  constant  magnetic  field,  a  sine  wave  of  e.m.f.  is 
generated.     In  a  circuit  with  constant  resistance  and  constant 
reactance,  this  sine  wave  of  e.m.f.  produces  a  sine  wave  of 
current.     Thus   distortion   of  the   wave-shape   or   higher   har- 
monics may  be  due  to  lack  of  uniformity  of  the  velocity  of 
the  revolving  conductor;  lack  of  uniformity  or  pulsation  of  the 
magnetic  field;  pulsation  of  the  resistance  or  pulsation  of  the 
reactance. 

341 


342         ALTERNATING-CURRENT  PHENOMENA 

The  first  two  cases,  lack  of  uniformity  of  the  rotation  or  of  the 
magnetic  field,  cause  higher  harmonics  of  e.m.f.  at  open  circuit. 
The  last,  pulsation  of  resistance  and  reactance,  causes  higher  har- 
monics only  when  there  is  current  in  the  circuit,  that  is,  underload. 

Lack  of  uniformity  of  the  rotation  is  hardly  ever  of  practical 
interest  as  a  cause  of  distortion,  since  in  alternators,  due  to 
mechanical  momentum,  the  speed  is  always  very  nearly  uniform 
during  the  period.  A  periodic  pulsation  of  speed  may  occur  in 
low  speed  singlephase  machines. 

Thus  as  causes  of  higher  harmonics  remain: 

1st.  Lack  of  uniformity  and  pulsation  of  the  magnetic  field, 
causing  a  distortion  of  the  generated  e.m.f.  at  open  circuit  as 
well  as  under  load. 

2d.  Pulsation  of  the  reactance,  causing  higher  harmonics  under 
load. 

3d.  Pulsation  of  the  resistance,  causing  higher  harmonics  under 
load  also. 

Taking  up  the  different  causes  of  higher  harmonics,  we  have : 

Lack  of   Uniformity  and  Pulsation  of  the  Magnetic  Field. 

234.  Since  most  of  the  alternating-current  generators  con- 
tain definite  and  sharply  defined  field-poles  covering  in  different 
types  different  proportions  of  the  pitch,  in  general  the  mag- 
netic flux  interlinked  with  the  armature  coil  will  not  vary  as  a 
sine  wave,  of  the  form 

$  cos  /3, 

but  as  a  complex  harmonic  function,  depending  on  the  shape 
and  the  pitch  of  the  field-poles  and  the  arrangement  of  the 
armature  conductors.  In  this  case  the  magnetic  flux  issuing 
from  the  field-pole  of  the  alternator  can  be  represented  by  the 
general  equation, 

$  =  AQ  +  Ai  cos/3  +  A2  cos  2  0  +  A3  cos  3  /8  -j-   .    ... 
+  £1  sin  0  +  £2  sin  2  ft  +  B3  sin  3  ft  +    .    .    ; 

If  the  reluctance  of  the  armature  is  uniform  in  all  directions, 
so  that  the  distribution  of  the  magnetic  flux  at  the  field-pole 
face  does  not  change  by  the  rotation  of  the  armature,  the  rate 
of  cutting  magnetic  flux  by  an  armature  conductor  is  <f>,  and 
the  e.m.f.  generated  in  the  conductor  thus  equal  thereto  in 
wave-shape.  As  a  rule  A0,  A2,  A4  .  .  .  B2,  B4  equal  zero; 
that  is,  successive  field-poles  are  equal  in  strength  and  distribu- 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   343 


tion  of  magnetism,  but  of  opposite  polarity.  In  some  types  of 
machines,  however,  especially  inductor  alternators,  this  is  not 
the  case. 

The  e.m.f.  generated  in  a  full-pitch  armature  turn  —  that  is, 
armature  conductor  and  return  conductor  distant  from  former 
by  the  pitch  of  the  armature  pole  (corresponding  to  the  distance 
from  field-pole  center  to  pole  center)  —  is 


de 


=  <£o  —  ^iso 

=  2  {  Ai  cos  ]8  +  As  cos  3  0  +  A5  cos  5  0  +  .    .    . 
+  Bi  sin  0  +  Bz  sin  3  0  +  £5  sin  5  0  +  .    . 


N6  Load 


x,=  =  146.5 


7 


FIG.  172. 

Even  with  an  unsymmetrical  distribution  of  the  magnetic 
flux  in  the  air-gap,  the  e.m.f.  wave  generated  in  a  full-pitch 
armature  coil  is  symmetrical,  the  positive  and  negative  half- 
waves  equal,  and  correspond  to  the  mean  flux  distribution  of 
adjacent  poles.  With  fractional  pitch- windings — that  is,  wind- 
ings whose  turns  cover  less  than  the  armature  pole-pitch — 
the  generated  e.m.f.  can  be  unsymmetrical  with  unsymmetrical 
magnetic  field,  but  as  a  rule  is  symmetrical  also.  In  unitooth 
alternators  the  total  generated  e.m.f.  has  the  same  shape  as  that 
generated  in  a  single  turn. 

With  the  conductors  more  or  less  distributed  over  the  surface 
of  the  armature,  the  total  generated  e.m.f.  is  the  resultant  of 


344         ALTERNATING-CURRENT  PHENOMENA 


several  e.m.fs.  of  different  phases,  and  is  thus  more  uniformly 
varying;  that  is,  more  sinusoidal,  approaching  sine  shape  to 
within  3  per  cent,  or  less,  as  for  instance  the  curves  Fig.  172 
and  Fig.  173  show,  which  represent  the  no-load  and  full-load 
wave  of  e.m.f .  of  a  three-phase  multitooth  alternator.  The  prin- 
cipal term  of  these  harmonics  is  the  third  harmonic,  which  con- 
sequently appears  more  or  less  in  all  alternator  waves.  As  a 
rule  these  harmonics  can  be  considered  together  with  the  har- 
monics due  to  the  varying  reluctance  of  the  magnetic  circuit. 

In  iron-clad  alternators  with  few  slots  and  teeth  per  pole,  the 
passage  of  slots  across  the  field-poles  causes  a  pulsation  of  the 


130 

W 

th  L 

oad 

120 

', 

-12 

7.0 

*= 

3.2 

/, 

-— 

^•^ 

>x 

110 

,/ 

\ 

100 

/ 

\ 

. 

90 

, 

/ 

V 

80 

/ 

\ 

70 

/ 

\\ 

60 

/ 

A 

.50 

2 

\ 

to 

f 

\ 

k 

30 

/ 

\ 

20 

/ 

\ 

10 

// 

\ 

0 

// 

^ 

-^, 

^ 

—  -v 

10 

0 

10 

20 

30 

to 

50 

60 

70 

80 

00 

100 

110 

120 

130 

ito 

150 

160 

170 

180 

FIG.  173. 

magnetic  reluctance,  or  its  reciprocal,  the  magnetic  reactance 
of  the  circuit.  In  consequence  thereof  the  magnetism  per  field- 
pole,  or  at  least  that  part  of  the  magnetism  passing  through  the 
armature,  will  pulsate  with  a  frequency  2  7,  if  7  =  number  of 
slots  per  pole. 

Thus,  in  a  machine  with  one  slot  per  pole  the  instantaneous 
magnetic  flux  interlinked  with  the  armature  conductors  can  be 
expressed  by  the  equation 


where 
and 


0  =  $  cos  /?  { 1  +  €  cos  [2  /3 
$  =  average  magnetic  flux, 
c  =  amplitude  of  pulsation, 
6  =  phase  of  pulsation. 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   345 

In  a  machine  with  7  slots  per  pole,  the  instantaneous  flux  inter- 
linked with  the  armature  conductors  will  be 

</»  =  $  cos  0  { 1  +  e  cos  [2  7/3  -  0] } , 

if  the  assumption  is  made  that  the  pulsation  of  the  magnetic 
flux  follows  a  simple  sine  law,  as  first  approximation. 

Irr  general  the  instantaneous  magnetic  flux  interlinked  with 
the  armature  conductors  will  be 

0  =  $  cos  0  { 1  +  ei  cos  (2  /?  -  00  +  €2  cos  (4  /3  -  02)  +  .    .    .  } , 

where  the  term  ey  is  predominating,  if  7  =  number  of  armature 
slots  per  pole.  This  general  equation  includes  also  the  effect  of 
lack  of  uniformity  of  the  magnetic  flux. 

In  case  of  a  pulsation  of  the  magnetic  flux  with  the  fre- 
quency, 2  7,  due  to  an  existence  of  7  slots  per  pole  in  the  arma- 
ture, the  instantaneous  value  of  magnetism  interlinked  with  the 
armature  coil  is 

0  =  3>  cos  0  { 1  +  e  cos  [2  7/3  -  0] } . 
Hence  the  e.m.f.  generated  thereby, 
dd> 

e"     ~nTt 

-  V2*f*jp  {cos  0(1  +  e  cos  [2  T£  -  0])}. 
And,  expanded, 

e  =  V2irfn$>  {  sin  ft  +  e  272~  *  sin  [  (2  7  -  1)0  -  0] 

+  e^^  sin  [(27+1)0-01  }' 

Hence,  the  pulsation  of  the  magnetic  flux  with  the  frequency, 
2  7,  as  due  to  the  existence  of  7  slots  per  pole,  introduces  two 
harmonics,  of  the  orders  (2  7  —  1)  and  (27  +  1). 

236.  If  7  =  1  it  is 

e  =  v/2  7r/n$  {  sin  ft  + 1  sin  (0  -  0)  +  ~  sin  (3  0  -  0)  1 ; 

I  Z  .4  J 

that  is,  in  a  unitooth  single-phaser  a  pronounced  triple  har- 
monic may  be  expected,  but  no  pronounced  higher  harmonics. 

Fig.  174  shows  the  wave  of  e.m.f.  of  the  main  coil  of  a  mono- 
cyclic  alternator  at  no  load,  represented  by, 


346         ALTERNATING-CURRENT  PHENOMENA 


e  =  E  {  sin  ft  -  0.242  sin  (30-  6.3)  -  0.046  sin  (5  0  -  2.6) 
+  0.068  sin  (7  0  -  3.3)  -  0.027  sin  (90-  10.0)  -  0.018 
sin  (11  0  -  6.6)  +  0.029  sin  (13  0  -  8.2)}; 

hence  giving  a  pronounced  triple  harmonic  only,  as  expected. 
If  7  =  2,  it  is, 

e  =  V2irfn3>  { sin  0  +  y  sin  (3  0  -  0)  +  y  sin  (5  0  -  0) }  , 

the  no-load  wave  of  a  unitooth  quarter-phase  machine,  having 
pronounced  triple  and  quintuple  harmonics. 


120 
110 
100 
90 
80 
70 
60 
50 
40 
30 
20 
10 
0 
-10 

I 

N  ^ 

^ 

•T^ 

^ 

\m 

^ 

V\ 

\w 

\ 

\r 

' 

\ 

\ 

\ 

Ji 

\ 

\ 

A 

? 

^ 

\ 

£ 

N 

/' 

D 

^ 

* 

X 

I 

<& 

S 

^ 

^ 

*? 

f 

^ 

\ 

^ 

-~^ 

3^ 

y 

, 

\ 

/ 

\ 

V 

^teo* 

"\ 

^/ 

1  — 

_  ** 

s 

"*«  — 

s 

0     10°    20'    30°   40°   50°    60°    70°  80°    90°   100°    110°   120°  130°1400  150  160°  170°     180° 
FIG.  174.  —  No-load  of  e.m.f.  of  unitooth  monocyclic  alternator. 

f  7  =  3,  it  is, 

e  = 


-  61) 


-^sin(7  0  -  0)1 

z  J 


That  is,  in  a  unitooth  three-phaser,  a  pronounced  quintuple 
and  septuple  harmonic  may  be  expected,  but  no  pronounced 
triple  harmonic. 

Fig.  175  shows  the  wave  of  e.m.f.  of  a  unitooth  three-phaser 
at  no-load,  represented  by 

e  =  E  {  sin  0  -  0.12  sin  (3  0  -  2.3)  -  0,23  sin  (5  0  -  1.5) 
+  0.134  sin  (7  0  -  6.2)  -  0.002  sin  (90  +  27.7)  -  0.046 
sin  (11  0  -  5.5)  +  0.031  sin  (13  0  -  61.5)). 

Thus   giving   a   pronounced   quintuple   and   septuple   and   a 


DISTORTION  OF  WAVE-SHAPE  AND  ITS  CA  USES   347 


lesser  triple  harmonic,  probably  due  to  the  deviation  of  the  field 
from  uniformity,  as  explained  above,  and  deviation  of  the 
pulsation  of  reluctance  from  sine-shape.  In  some  especially 
favorable  cases,  harmonics  as  high  as  the  35th  and  37th  have  been 
observed,  caused  by  pulsation  of  the  reluctance,  and  even  still 
higher  harmonics. 

In  general,  if  the  pulsation  of  the  magnetic  reactance  is  denoted 
by  the  general  expression 

00 

1  +  2  767r  cos  (2  T|3  -  07), 


0    10    20°       30    40°  50°    60°    70°  80°    90°   100°  110°    120°  130  1.40*150160 170 180 
FIG.  175. — No-load  wave  of  e.m.f.  of  unitooth  three-phase  alternator. 

the  instantaneous  magnetic  flux  is 


=  3>  cos  - 


7  cos  (2  70  -  0T) 


cos  0  +      cos  (ft  -  00  + 


°°  r 

2v    £ 

i    L2 


cos  [(2  7 


n      COS  1(2  T 


hence,  the  e.m.f., 


sn 


sn    0  - 


[e7  sin  [(2  T  +  D0  -  *T]  +  €T+1  sin  [(2  7  +  1)0  -  07+J] 


348         ALTERNATING-CURRENT  PHENOMENA 

With  the  general  adoption  of  distributed  fractional  pitch  arma- 
ture windings,  such  pronounced  wave  shape  distortions  as  shown 
by  the  unitooth  alternators  shown  as  illustrations,  have  become 
infrequent. 

Pulsation  of  Reactance 

236.  The  main  causes  of  a  pulsation  of  reactance  are  mag- 
netic saturation  and  hysteresis,  and  synchronous  motion.     Since 
in  an  iron-clad  magnetic  circuit  the  magnetism  is  not  propor- 
tional to  the  m.m.f.,  the  wave  of  magnetism  and  thus  the  wave 
of  e.m.f.  will  differ  from  the  wave  of  current.     As  far  as  this 
distortion  is  due  to  the  variation  of  permeability,  the  distortion  is 
symmetrical  and  the  wave  of  generated  e.m.f.  represents  no 
power.     The  distortion  caused  by  hysteresis,  or  the  lag  of  the 
magnetism  behind  the  m.m.f.,  causes  an  unsymmetrical  distor- 
tion of  the  wave  which  makes  the  wave  of  generated   e.m.f. 
differ  by  more  than  90°  from  the  current  wave  and   thereby 
represents  power — the  power  consumed  by  hysteresis. 

In  practice  both  effects  are  always  superimposed;  that  is, 
in  a  ferric  inductive  reactance,  a  distortion  of  wave-shape  takes 
place  due  to  the  lack  of  proportionality  between  magnetism  and 
m.m.f.  as  expressed  by  the  variation  in  the  hysteretic  cycle. 

This  pulsation  of  reactance  gives  rise  to  a  distortion  con- 
sisting mainly  of  a  triple  harmonic.  Such  current  waves  dis- 
torted by  hysteresis,  with  a  sine  wave  of  impressed  e.m.f.,  are 
shown  in  Figs.  80  and  81,  Chapter  XII,  on  Hysteresis.  In- 
versely, if  the  current  is  a  sine  wave,  the  magnetism  and  the 
e.m.f.  will  differ  from  sine-shape. 

For  further  discussion  of  this  distortion  of  wave-shape  by 
hysteresis,  Chapter  XII  may  be  consulted. 

237.  Distortion  of  wave-shape  takes  place  also  by  the  pul- 
sation of  reactance  due  to  synchronous  rotation,  as   discussed 
in  the  chapter  on  Reaction  Machines,  in  "Theory  and  Calculation 
of  Electrical  Apparatus." 

With  a  sine  wave  of  e.m.f.,  distorted  current  waves  result. 
Inversely,  if  a  sine  wave  of  current, 

i  =  I  cos  0, 

exists  through  a  circuit  of  synchronously  varying  reactance, 
as  for  instance,  the  armature  of  a  unitooth  alternator  or  syn- 
chronous motor — or,  more  general,  an  alternator  whose  arma- 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   349 

ture  reluctance  is  different  in  different  positions  with  regard  to 
the  field-poles — and  the  reactance  is  expressed  by 

X  =  x  {1  +  €cos  (20  -  61)}; 

or,  more  general, 

r 

X  =  x     1  +    2yey  cos  (2  70  -  By 
i 

the  wave  of  magnetism  is 
X  x     ! 

0   =   ^— r  COS  0   =  — — r-      COS  0  -f-    2LeY  COS  0  COS  (270  —   0T) 

Z  Trjn.  2  irjn  {  i 


cos  0  +  i1  cos  (0  -  00  +  ST  I  ^  cos  [(2  7  +  1) 

^  i 


[(2.T 


hence  the  wave  of  generated  e.m.f., 


'^5(8 

€l 

=  x  i  sin  0  +  —  sin  (0  —  0i)  + 
€2  y  ,  sin  [(2  7  +  1)0 


that  is,  the  pulsation  of  reactance  of  frequency,  2  7,  introduces 
two  higher  harmonics  of  the  order  (2  7  —  1)  and  (27  +  !). 
If 

X  =  x{l  +c  cos  (20  -  0)}, 
it  is 


cos/?  +     cos  ^  ~~ 


e  =  a;     sn 


sn    8  - 


cos  3  ^  ~ 


-sm  (3  j8  - 


Since  the  pulsation  of  reactance  due  to  magnetic  saturation 
and  hysteresis  is  essentially  of  the  frequency,  2  /  —  that  is, 
describes  a  complete  cycle  for  each  half-wave  of  current  —  this 
shows  why  the  distortion  of  wave-shape  by  hysteresis  consists 
essentially  of  a  triple  harmonic. 

The  phase  displacement  between  e  and  i,  and  thus  the  power 
consumed  or  produced  in  the  electric  circuit,  depends  upon  the 
angle,  9,  as  discussed  before. 


350         ALTERNATING-CURRENT  PHENOMENA 

238.  In  case  of  a  distortion  of  the  wave-shape  by  reactance, 
the  distorted  waves  can  be  replaced  by  their  equivalent  sine 
waves,  and  the  investigation  with  sufficient  exactness  for  most 
cases  be  carried  out  under  the  assumption  of  sine  waves,  as  done 
in  the  preceding  chapters. 

Similar  phenomena  take  place  in  circuits  containing  polari- 
zation cells,  leaky  condensers,  or  other  apparatus  representing 
a  synchronously  varying  negative  reactance.  Possibly  dielectric 
hysteresis  in  condensers  causes  a  distortion  similar  to  that  due 
to  magnetic  hysteresis. 

Inversely,  at  very  high  voltages,  where  corona  appears  on 
the  conductors,  with  a  sine  wave  of  impressed  voltage,  a  distor- 
tion of  the  capacity  current  wave  occurs,  which  is  largely  effect- 
ive, but  partly  reactive  due  to  the  increase  of  capacity  under 
corona. 

Pulsation  of  Resistance 

239.  To  a  certain  extent  the  investigation  of  the  effect  of 
synchronous  pulsation  of  the  resistance  coincides  with  that  of 
reactance;  since  a  pulsation  of  reactance,  when  unsymmetrical 
with  regard  to  the  current  wave,  introduces  a  power  component 
which  can  be  represented  by  an  "effective  resistance." 

Inversely,  an  unsymmetrical  pulsation  of  the  ohmic  resistance 
introduces  a  wattless  component,  to  be  denoted  by  "effective 
reactance." 

A  typical  case  of  a  synchronously  pulsating  resistance  is 
represented  in  the  alternating  arc. 

The  apparent  resistance  of  an  arc  depends  upon  the  current 
through  the  arc;  that  is,  the  apparent  resistance  of  the  arc  = 

potential  difference  between  electrodes  .    ,  .  ,   -  ,, 

is  high  for  small  currents, 

current 

low  for  large  currents.  Thus  in  an  alternating  arc  the  apparent 
resistance  will  vary  during  every  half-wave  of  current  between  a 
maximum  value  at  zero  current  and  a  minimum  value  at  maxi- 
mum current,  thereby  describing  a  complete  cycle  per  half-wave 
of  current. 

Let  the  effective  value  of  current  through  the  arc  be  repre- 
sented by  /. 

Then  the  instantaneous  value  of  current,  assuming  the  current 
wave  as  sine  wave,  is  represented  by 

i  =  I  V2  sin  0; 


DISTORTION  OF  WAVE-SHAPE  AND  ITS  CA  USES   351 

and  the  apparent  resistance  of  the  arc,  in  first  approximation,  by 

R  =  r(l  +€0082)3); 
thus  the  potential  difference  at  the  arc  is 

e  =  iR  =  I\/2r  sin  ft  (1  +  e  cos  2  /?) 

=  rl \/2  {  ( 1  -  I)  sin  ft  +  J-  sin  3 
Hence  the  effective  value  of  potential  difference, 


-r/^I 


and  the  apparent  resistance  of  the  arc, 

r0  =  j-  = 
The  instantaneous  power  consumed  in  the  arc  is 

ie  =  2  rl2 1  (l  -  ^}  sin2  ft  +  ^  sin  /3  sin  3  0 ) 

I    \  Z/  A  J 

Hence  the  effective  power, 
P  =  1 

The  apparent  power,  or  volt-amperes  consumed  by  the  arc, 


-c-f     - 
Thus  the  power-factor  of  the  arc, 


that  is,  less  than  unity. 

240.  We  find  here  a  case  of  a  circuit  in  which  the  power-factor 
— that  is,  the  ratio  of  watts  to  volt-amperes — differs  from  unity 
without  any  displacement  of  phase;  that  is,  while  current  and 
e.m.f.  are  in  phase  with  each  other,  but  are  distorted,  the  alter- 
nating wave  cannot  be  replaced  by  an  equivalent  sine  wave, 


352         ALTERNATING-CURRENT  PHENOMENA 


since  the  assumption  of  equivalent  sine  wave  would  introduce  a 
phase  displacement, 

cos  9  =  p 

of  an  angle,  6,  whose  sign  is  indefinite. 

As  an  example  are  shown,  in  Fig.  176,  for  the  constants,  /  =  12, 
r  =  3,  e  =  0.9,  the  resistance, 

R  =  3  (1  +  0.9  cos  2/8); 
the  current, 

i  =  17  sin  0; 


\ 


AF  I  ABLE 


=  28( 


1  +.  9  cis  2  6) 


RESISTANCE 


3/3) 


A 


FIG.  176. — Periodically  varying  resistance. 
the  potential  difference, 

e  =  28  (sin  /3  +  0.82  sin  3  /3). 
In  this  case  the  effective  e.m.f.  is 

E  =  25.5; 
the  apparent  resistance, 

r0  =  2.13; 
the  power, 

P  =  244; 
the  apparent  power, 

El  =  307; 
the  power-factor, 

p  =  0.796. 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES  353 


As  seen,  with  a  sine  wave  of  current  the  e.m.f.  wave  in  an 
alternating  arc  will  become  double-peaked,  and  rise  very  abruptly 
near  the  zero  values  of  current.  Inversely,  with  a  sine  wave  of 
e.m.f.  the  current  wave  in  an  alternating  arc  will  become  peaked, 
and  very  flat  near  the  zero  values  of  e.m.f. 

241.  In  reality  the  distortion  is  of  more  complex  nature, 
since  the  pulsation  of  resistance  in  the  arc  does  not  follow  a 
simple  sine  law  of  double  frequency,  but  varies  much  more 
abruptly  near  the  zero  value  of  current,  making  thereby  the 
variation  of  e.m.f.  near  the  zero  value  of  current  much  more 
abruptly,  or,  inversely,  the  variation  of  current  more  flat. 


10   11   13    13   14    15^/16   17 


ONE  PAIR  CARBONS 
REGULATED  BY  HANDJT\ 

I..  A,  C.  dynamo  e,  m-,  f,||  \ 
II.  *-f  '"        "      currents 
III."  '"       "      watts.  , 

V 


21  23  X 


FIG.  177. — Electric  arc. 


A  typical  wave  of  potential  difference,  with  an  approximate 
sine  wave  of  current  through  the  arc,  is  given  in  Fig.  177.1 

242.  The  value  of  e,  the  amplitude  of  the  resistance  pulsation, 
largely  depends  upon  the  nature  of  the  electrodes  and  the 
steadiness  of  the  arc,  and  with  soft  carbons  and  a  steady  arc  is 
small,  and  the  power-factor,  p,  of  the  arc  near  unity.  With  hard 
carbons  and  an  unsteady  arc,  e  rises  greatly,  higher  harmonics 
appear  in  the  pulsation  of  resistance,  and  the  power-factor,  p, 
falls,  being  in  extreme  cases  even  as  low  as  0.6.  Especially  is 
this  the  case  with  metal  arcs. 

This  double-peaked  appearance  of  the  voltage  wave,  as  shown 
by  Figs.  176  and  177,  is  characteristic  of  the  arc  to  such  an  extent 

1  From  American  Institute  of  Electrical  Engineers,  Transactions,  1890, 
p.  376.     Tobey  and  Walbridge,  on  the  Stanley  Alternate  Arc  Dynamo. 
23 


354         ALTERNATING-CURRENT  PHENOMENA 

that  when  in  the  investigation  of  an  electric  circuit  by  oscillo- 
graph such  a  wave-shape  is  found,  the  existence  of  an  arc  or 
arcing  ground  somewhere  in  the  circuit  may  usually  be  sus- 
pected. This  is  of  importance  as  in  high- voltage  systems  arcs 
are  liable  to  cause  dangerous  voltages. 

The  pulsation  of  the  resistance  in  an  arc,  as  shown  in  Fig.  177 
for  hard  carbons,  is  usually  very  far  from  sinusoidal,  as  assumed 
in  Fig.  176.  It  is  due  to  the  feature  of  the  arc  that  the  voltage 
consumed  in  the  arc  flame  decreases  with  increase  of  current — 
approximately  inversely  proportional  to  the  square  root  of  the 
current — and  so  is  lowest  at  maximum  current. 

Approximately,  the  volt-ampere  characteristic  of  the  arc  can 
be  represented  by, 

s* 

e  =  CQ  +  — j=,  (1) 

where  eo  is  a  constant  of  the  electrode  material  (mainly),  c  a  con- 
stant depending  also  upon  the  electrode  material  and  on  the 
arc  length,  and  approximately  proportional  thereto. 

This  equation  would  give  e  =  <»,  for  i  =  0.  This  obviously 
is  not  feasible.  However,  besides  the  arc  conduction  as  given 
by  above  equation — which  depends  upon  mechanical  motion 
of  the  vapor  stream — a  slight  conduction  also  takes  place 
through  the  residual  vapor  between  the  electrodes,  as  a  path  of 
high  resistance,  r,  and  near  zero  current,  where  the  voltage  is 
not  sufficient  to  maintain  an  arc,  this  latter  conduction  carries 
the  current. 

The  characteristic  of  the  alternating-current  arc  therefore 
consists  of  the  combination  of  two  curves :  the  arc  characteristic, 
(1),  and  the  resistance  characteristic, 

e  =  ri.  (2) 

The  phenomenon  then  follows  that  curve  which  gives  the 
lowest  voltage;  that  is,  for  high  values  of  current,  is  represented 
by  equation  (1),  for  low  values  of  current,  by  equation  (2). 

243.  As  an  example  are  shown  in  Fig.  178  the  calculated 
curves  of  an  alternating  arc  between  hard  carbons  (or  carbides) , 
for  the  constants,' 

€Q   =  30  VoltS, 

c   =  40, 

r   =  70  ohms. 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   355 

The  curve  I  represents  the  arc  conduction,  following  equation 

(1), 

e  =  30  +  -~L 
Vt 

and  the  curve  II  represents  the  conduction  through  the  (sta- 
tionary) residual  vapor,  by  equation  (2),  near  the  zero  points, 
A  and  D}  of  the  current, 

e  =  70  i. 

As  seen,  from  A  to  B  the  voltage  varies  approximately  pro- 
portionally with  the  current.     At  B  the  arc  starts,  and  the  vol- 


\]D 


FIG.  178. 

tage  drops  with  the  further  increase  of  current,  and  then  rises 
again  with  the  decreasing  current,  until  at  C,  the  intersection 
point  between  curves  I  and  II,  the  arc  extinguishes  and  the 
voltage  follows  curve  II,  until  at  E  the  arc  starts  again.  The 
two  sharp  peaks  of  the  curve  thus  represent  respectively  the 
starting  and  the  extinction  of  the  arc. 

Since  the  high  values  of  voltage  near  zero  current  lower  and  the 
low  values  of  voltage  near  maximum  current  raise  the  value  of 


356         ALTERNATING-CURRENT  PHENOMENA 

the  current,  the  current  wave  does  not  remain  a  sine  wave,  if 
the  arc  voltage  is  an  appreciable  part  of  the  total  voltage,  but 
the  current  wave  becomes  peaked,  with  flat  zero,  as  expressed 
approximately  by  a  third  harmonic  in  phase  with  the  funda- 
mental. The  current  wave  in  Fig.  178  so  has  been  assumed  as 

i  =  13  cos  0  +  2  cos  3  0. 

From  Fig.  178  follows: 

effective  value  of  current,       9.30  amp., 
effective  value  of  voltage,     47.2  volts; 

hence,  volt-amperes  consumed  by  the  arc,  439  volt-amp.;  and, 
by  averaging  the  products  of  the  instantaneous  values  of  volts 
and  amperes, 

power  consumed  in  the  arc,         388  watts; 
hence, 

power-factor,  77  per  cent. 

If  the  resistance,  r,  of  the  residual  arc-vapor  is  lower,  as  by 
the  use  of  softer  carbons,  for  instance,  given  by 

r  =  30  ohms, 

as  shown  by  the  dotted  curve,  II',  in  Fig.  178,  the  voltage  peaks 

are  greatly  cut  down,  giving  a  lesser  wave-shape  distortion,  and 

so, 

effective  value  of  voltage,        43.1  volts, 
volt-amperes  in  arc,  395  volt-amp., 

watts  in  arc,  335  watts, 

hence, 

power-factor,  85  per  cent. 

Comparing  Fig.  178  with  177  shows  that  178  fairly  well  approxi- 
mates 177,  except  that  in  Fig.  177  the  second  peak  is  lower  than 
the  first.  This  is  due  to  the  lower  resistance,  r,  of  the  residual 
vapor  immediately  after  the  passage  of  the  arc  than  before  the 
starting  of  the  arc.  Fig.  177  also  shows  a  decrease  of  resistance, 
r,  immediately  before  starting,  or  after  extinction  of  the  arc, 
which  may  be  represented  by  some  expression  like 

r  =  r0i-b, 
where  b  <  1, 

but  which  has  not  been  considered  in  Fig.  178. 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES  357 

The  softer  the  carbons,  the  more  is  the  latter  effect  appreciable 
and  the  peaks  rounded  off,  thus  causing  the  curve  to  approach 
the  appearance  of  Fig.  176,  while  with  metal  arcs,  where  r  is 
very  high,  the  peaks,  especially  the  first,  become  very  sharp 
and  high,  frequently  reaching  values  of  several  thousand  volts. 

Further  discussion  on  the  effect  of  the  arc  see  "Theory  and 
Calculation  of  Electric  Circuits." 

244.  One  of  the  most  important  sources  of  wave-shape  dis- 
tortion is  the  presence  of  iron  in  a  magnetic  circuit.  The  mag- 
netic induction  in  iron,  and  therewith  the  magnetic  flux,  is  not 
proportional  to  the  magnetizing  force  or  the  exciting  current, 
but  the  magnetic  induction  and  the  magnetizing  force  are  related 
to  each  other  by  the  hysteresis  cycle  of  the  iron,  as  discussed  in 
Chapter  XII.  In  an  iron-clad  magnetic  circuit,  the  magnetic 


FIG.  179. 

flux  and  the  current,  therefore,  cannot  both  be  sine  waves;  if  the 
magnetic  flux  and  therefore  the  generated  e.m.f.  are  sine  waves, 
the  current  \liffers  from  sine  wave-shape,  while  if  a  sine  wave  of 
current  is  sent  through  the  circuit,  the  magnetic  flux  and  the 
generated  e.m.f.  cannot  be  sine  waves. 

A.  Sine  Wave  of  Voltage 

Let  a  sine  wave  of  e.m.f.  be  impressed  upon  an  iron-clad 
reactance  coil,  or  a  primary  coil  of  a  transformer  with  open 
secondary  circuit.  Neglecting  the  ohmic  resistance  of  the 
circuit,  that  is,  assuming  the  generated  e.m.f.  as  equal  or 
practically  equal  to  the  impressed  e.m.f.,  the  voltage  consumed 
by  the  generated  e.m.f.  and  therewith  the  magnetic  flux  are 
sine  waves,  as  represented  by  E  and  B  in  Fig.  179.  The  cur- 


358         ALTERNATING-CURRENT  PHENOMENA 

rent  which  produces  this  magnetic  flux,  B,  and  so  the  voltage, 
E,  then  is  derived  point  by  point  from  B,  by  the  hysteresis  cycle 
of  the  iron.  With  the  hysteresis  cycle  given  in  Fig.  180,  the 
current  then  has  the  wave-shape  given  as  /  in  Fig.  179,  that  is, 
greatly  differs  from  a  sine  wave.  This  distortion  of  the  current 
wave  is  mainly  due  to  the  bend  of  the  magnetic  characteristic, 
that  is,  the  magnetic  saturation,  and  not  to  the  energy  loss  or 
the  area  of  the  curve.  This  is  seen  by  resolving  the  current  wave, 
/,  into  two  components:  an  energy  component,  i',  in  phase  with 


FIG.  180. 

the  e.m.f.,  e  =  E  sin  <f>,  and  a  wattless  component,  i",  in  quadra- 
ture with  E,  and  in  phase  with  B.  These  components  are  calcu- 
lated as 


and 


t    - 


where  i+  and  &V-0  are  the  instantaneous  values  of  the  current,  I, 
at  the  angles  <f>  and  w  —  <j>,  respectively. 

These  components,  the  hysteresis  power  current,  i',  and  the 
reactive  magnetizing  current,  i"  ,  are  plotted  in  Fig.  181  and 
show  that  i'  is  nearly  a  sine  wave,  while  i"  is  greatly  distorted 
and  peaked. 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   359 

The  total  current,  /,  derived  by  the  hysteresis  cycle,  Fig.  180, 
from  the  magnetic  flux, 

B    =   BQ  COS  0, 

can  be  resolved  into  an  infinite  series  of  harmonic  waves,  that  is, 
a  trigonometric  or  Fourier  series  of  the  form: 

i  =  a\  cos  0  +  as  cos  3  0  +  a&  cos  50+.    .    .  +  an  cos  n<f>  +  ,   .    . 
+  61  sin  0  +  63  sin  3  0  +  65  sin  5  0  +.    .    .  +  bn  sin  n0  +  .    .    . 

or  of  the  form : 

i  =  GI  cos  (0  —  0i)  +  c3  cos  (3  0  —  03)  +  c5  cos  (5  0  —  06) 
+  ,        .  +  cn  cos  (nct>  —  On)  +  . 


where 


FIG.  181. 


tan  er 


The  coefficients  an  and    6n  are  determined    by  the  definite 
integrals:1 

2       . 

*  cos  n<f>d<j>  =  2  X  ct^g      cos 


2  /•' 

6n  =  -  I    i  sin  ?^0d0  =  2  X  avg  (^sinn0)0ir; 
TT»/O 

that  is,  by  multiplying  the  instantaneous  values  of  i,  as  given 
numerically,  by  cos  n0  and  sin  n0,  respectively,  and  then 
averaging. 

JSee  "Engineering  Mathematics." 


360         ALTERNATING-CURRENT  PHENOMENA 

Just  as  in  most  investigations  dealing  with  alternating  currents, 
not  the  fundamental  sine  wave,  but  the  fundamental  sine  wave 
together  with  all  its  higher  harmonics,  that  is,  the  total  wave,  is 
of  importance;  so  also  when  dealing  with  the  higher  harmonics, 
frequently  not  the  individual  higher  harmonic  sine  wave  is  of 
importance,  but  the  higher  harmonic  together  with  all  of  its 
higher  harmonics.  For  instance,  when  dealing  with  the  disturb- 
ances caused  by  the  third  harmonic  in  a  three-phase  system,  the 
third  harmonic  together  with  all  its  higher  harmonics  or  over- 
tones, as  the  ninth,  fifteenth,  twenty-first,  etc.,  comes  in  consid- 
eration, that  is,  all  the  components  which  repeat  after  one-third 
cycle.  The  higher  harmonic  then  appears  as  a  distorted  wave, 
including  its  higher  harmonics. 

To  determine,  from  the  instantaneous  values  of  a  distorted 
wave,  the  instantaneous  values  of  its  nth  harmonic  distorted 
wave,  that  is,  the  nth  harmonic  together  with  its  overtones,  of 
order  3  n,  5  n,  7  n,  etc.,  the  average  is  taken  of  n  instantaneous 
values  of  the  total  wave  (or  any  component  thereof,  which 
includes  the  nth  harmonic),  differing  from  each  other  in  phase  by 

-  period.     That  is,  it  is 

n-l 


This  method  is  based  on  the  relations: 

n~l          i         ,    2inr\ 

2  K  cos  I  md>  H 1  =  n  cos 

V  n  / 


1  /  2/C7T\ 

*  sm  I  m(f)  H )  =  n  sm  m<f>, 

\  n  I 


if  m  =  n  or  if  m  is  a  multiple  of  n;  otherwise  these  sums  =  0, 
where  m  and  n  are  integer  numbers. 

245.  In  .this  manner  the  wave  of  exciting  current,  7,  of  Fig. 
179  is  resolved,  in  Fig.  182,  into  the  fundamental  sine  wave,  ii, 
and  the  higher  harmonics,  iz,  i$,  ii,  which  are  general  waves,  that 
is,  include  their  higher  harmonics. 

Analytically,  it  can  be  represented  by 

i  =  8.857  cos  (0  +  37.6°)  +  1.898  cos  3  (0  +  4.1°) 
+  0.585  cos  5  (0  -  1.7°)  +  0.319  cos  7  (0  -  3.2°) 
+  0.158  cos  9  (<£  -  2.5°)  +  .  .  • 

where  B  =  10,000  cos  <£  is  the  wave  of  magnetic  induction. 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   361 

The  equivalent  sine  wave  of  above  current  wave  is 
IQ  =  9.104  cos  (0  -  36.3°). 

In  this  case  of  the  distortion  of  a  current  wave  by  an  iron-clad 
reactance  coil  or  transformer,  with  a  sine  wave  of  impressed 
e.m.f.,  it  is,  from  the  above  equation  of  the  current  wave, 

Effective  value  of  the  total  current  .    .    .  ' .    .    .    .   6 . 423 
Effective  value  of  its  fundamental  sine  wave  .   .    .   6 . 27 
Effective  value  of  the  sum  of  all  its  higher  harmonics  1 . 43. 

That  is,  the  effective  value  of  all  the  harmonics  is  22.3  per  cent,  of 
the  effective  value  of  the  total  current. 


u 


nX 


\ 


V 


\\ 


\ 


^ 


FIG.  182. 


B.  Sine  Wave  of  Current 

246.  If  a  sine  wave  of  current  exists  through  an  iron-clad 
magnetic  circuit,  as,  for  instance,  an  iron-clad  reactance  coil  or 
transformer  connected  in  series  to  a  circuit  traversed  by  a  sine 
wave,  the  potential  difference  at  the  terminals  of  the  reactance 
cannot  be  a  sine  wave,  but  contains  higher  harmonics. 

From  the  sine  wave  of  current 

i  —  I  cos  0, 

follows  by  the  hysteresis  cycle,  Fig.  180,  the  wave  of  magnetism. 
This  is  not  a  sine  wave,  but  hollowed  out  on  the  rising,  humped 
on  the  decreasing  side,  that  is,  has  a  distortion  about  opposite 


362         ALTERNATING-CURRENT  PHENOMENA 


from  that  of  the  current  wave  in  Fig.  179;  the  wave  of  magnetism 
has  the  maximum  at  the  same  angle,  0,  as  the  current,  but  passes 
the  zero  much  later  than  the  current. 

From  the  wave  of  magnetism  follows  the  wave  of  generated 
e.m.f.,  and  so  (approximately,  that  is,  neglecting  resistance)  of 

terminal  voltage,  e,  at  the  reactance,  since  e  is  proportional  to  -Tr- 
Ct  0 

It  is  plotted  as  E  in  Fig.  183,  and  resolved  into  its  harmonics 
in  the  same  manner  as  the  current  wave  in  A. 


FIG.  183. 

As  seen,  with  a  sine  wave  of  current  traversing  an  iron-clad 
reactance,  the  e.m.f.  wave  is  very  greatly  distorted,  and  the 
maximum  value  of  the  distorted  e.m.f.  wave  is  more  than  twice 
the  maximum  of  its  fundamental  sine  wave. 

Denoting  the  current  wave  by, 

i  =  10  sin  (<f>  +  30°), 
the  e.m.f.  wave  in  Fig.  183  is  represented  by 

e  =  11.67  cos  (0  +  2.5°)  +  6.64  cos  3  (0  -  1.13°) 

+  3.24  cos  5  (0  -  2.4°)  +  1.8  cos  7  (0  -  1.53°)  + 

1.16  cos  9  (0  -  0.5°)  +  0.80  cos  11  (0  -  2°) 

+  0.53  cos  13  (0  -  2°)  +  0.19  cos  15  (0  -  1°)  +  .    .    . 

that  is,  all  the  harmonics  are  nearly  in  phase  with  each  other,  so 
accounting  for  the  very  steep  peak.     It  is 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES  363 

Effective  value  of  total  wave   ..........     9.91 

.Effective  value  of  its  fundamental  sine  wave    .    .    .     8  .  25 
Effective  value  of  the  sum  of  all  its  higher  harmonics    5  .  48 

that  is,  the  effective  value  of  all  the  higher  harmonics  is  55.3  per 
cent,  of  the  effective  value  of  the  total  wave. 

The  impedance  of  this  iron-clad  reactance,  with  a  sine  wave 
current  of  7.07  effective,  so  is 


while  ,the  same  reactance,    with    a   sine    wave   e.m.f.    of   7.07 
effective,  in  A,  gives  the  impedance, 


The  conclusion  is  that  an  iron-clad  magnetic  circuit  is  not 
suitable  for  a  reactor,  since  even  below  saturation  (as  above 
assumed)  it  produces  very  great  wave-shape  distortion. 

As  discussed  before,  the  insertion  of  even  a  small  air-gap  into 
the  magnetic  circuit  makes  the  current  wave  nearly  coincide  in 
phase  and  in  shape  with  the  wave  of  magnetism. 

C.  Three-phase  Circuits 

247.  The  wave-shape  distortion  in  an  iron-clad  magnetic 
circuit  has  an  important  bearing  on  transformer  connections  in 
three-phase  circuits. 

The  e.m.fs.  and  the  currents  in  a  three-phase  system  are  dis- 
placed from  each  other  in  phase  by  one-third  of  a  period  or  120°. 
Their  third  harmonics,  therefore,  differ  by  3  X  120°,  or  a  com- 
plete period,  that  is,  are  in  phase  with  each  other.  That  is,  what- 
ever third  harmonics  of  e.m.f.  and  of  current  may  exist  in  a 
three-phase  system  must  be  in  phase  with  each  other  in  all 
three  phases,  or,  in  other  words,  for  the  third  harmonics  the 
three-phase  system  is  single-phase. 

The  sum  of  the  three  e.m.fs.  between  the  lines  of  a  three-phase 
system  (A  voltages)  is  zero.  Since  their  third  harmonic  would 
be  in  phase  with  each  other,  and  so  add  up,  it  follows: 

The  voltages  between  the  lines  of  a  three-phase  system,  or  A 
voltages,  cannot  contain  any  third  harmonic  or  its  overtones 
(ninth,  fifteenth,  twenty-first,  etc.,  harmonics). 

Since  in  a  three-wire,  three-phase  system  the  sum  of  the  three 


364         ALTERNATING-CURRENT  PHENOMENA 

currents  in  the  line  is  zero,  but  their  third  harmonics  would  be  in 
phase  with  each  other,  and  their  sum,  therefore,  not  zero,  it 
follows : 

The  currents  in  the  lines  of  a  three-wire,  three-phase  system, 
or  Y  currents,  cannot  contain  any  third  harmonic. 

Third  harmonics,  however,  can  exist  in  the  Y  voltage  or  voltage 
between  line  and  neutral  of  the  system,  and  since  the  third  har- 
monics are  in  phase  with  each  other,  in  this  case,  a  potential 
difference  of  triple  frequency  exists  between  the  neutral  of  the 
system  and  all  three  phases  as  the  other  terminal,  that  is,  the 
whole  system  pulsates  against  the  neutral  at  triple  frequency. 

Third  harmonics  can  also  exist  in  the  currents  between  the 
lines,  or  A  currents.  Since  the  two  currents  from  one  line  to  the 
other  two  lines  are  displaced  60°  from  each  other,  their  third 
harmonics  are  in  opposition  and,  therefore,  neutralize.  That  is, 
the  third  harmonics  in  the  A  currents  of  a  three-phase  system 
do  not  exist  in  the  Y  currents  in  the  lines,  but  exist  only  in  a 
local  closed  circuit. 

Third  harmonics  can  exist  in  the  line  currents  in  a  four-wire, 
three-phase  system,  as  a  system  with  grounded  neutral.  In  this 
case  the  third  harmonics  of  currents  in  the  lines  return  jointly 
over  the  fourth  or  neutral  wire,  and  even  with  balanced  load  on 
the  three  phases,  the  neutral  wire  carries  a  current  which  is  of 
triple  frequency. 

248.  With  a  sine  wave  of  impressed  e.m.f.  the  current  in  an 
iron-clad  circuit,  as  the  exciting  current  of  a  transformer,  must 
contain  a  strong  third  harmonic,  otherwise  the  e.m.f.  cannot 
be  a  sine  wave.  Since  in  the  lines  of  a  three-phase  system  the 
third  harmonics  of  current  cannot  exist,  interesting  wave-shape 
distortions  thus  result  in  transformers,  when  connected  to  a  three- 
phase  system  in  such  a  manner  that  the  third  harmonic  of  the 
exciting  current  would  have  to  enter  the  line  as  Y  current,  and 
so  is  suppressed. 

For  instance,  connecting  three  iron-clad  reactors,  as  the 
primary  coils  of  three  transformers — with  their  secondaries 
open-circuited — in  star  or  Y  connection  into  a  three-phase 
system,  with  a  sine  wave  of  e.m.f.,  e,  impressed  upon  the  lines. 
Normally,  the  voltage  of  each  transformer  should  be  a  sine  wave 

/> 

also,  and  equal  — -j=-     This,  however,  would  require  that  the 
V  3 

current  taken  by  the  transformer  as  exciting  current  contains  a 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   365 

third  harmonic.  As  such  a  third  harmonic  cannot  exist  in  a 
three-phase  circuit,  the  wave  of  magnetism  cannot  be  a  sine  wave, 
but  must  contain  a  third  harmonic,  about  opposite  to  that  which 
was  suppressed  in  the  exciting  current.  The  e.m.f.  generated 
by  this  magnetism,  and  therewith  the  potential  difference  at 
the  transformer  or  Y  voltage,  therefore,  must  also  contain  a 
third  harmonic,  and  its  overtones,  three  times  as  great  as  that  of 
the  magnetism,  due  to  the  triple  frequency. 

With  three  transformers  connected  in  Y  into  a  three-phase 
system  with  open  secondary  circuit,  we  have,  then,  with  a  sine 
wave  of  e.m.f.  impressed  between  the  three-phase  lines,  the 
conditions: 

The  voltage  at  the  transformers,  or  Y  voltage,  cannot  be  a  sine 
wave,  but  must  contain  a  third  harmonic  and  its  overtones,  but 
can  contain  no  other  harmonics,  since  the  other  harmonics,  as 
the  fifth,  seventh,  etc.,  would  not  eliminate  by  combining  two 
Y  voltages  to  the  A  voltage  or  line  voltage,  and  the  latter  was 
assumed  as  sine  wave. 

The  exciting  current  in  the  transformers  cannot  contain  any 
third  harmonic  or  its  overtones,  but  can  contain  all  other 
harmonics. 

The  magnetic  flux  is  not  a  sine  wave,  but  contains  a  third 
harmonic  and  its  overtones,  corresponding  to  those  of  the  Y 
voltage,  but  contains  no  other  harmonics,  and  is  related  to  the 
exciting  current  by  the  hysteresis  cycle. 

Herefrom  then  the  wave-shapes  of  currents,  magnetism  and 
voltage  can  be  constructed.  Obviously,  since  the  relation 
between  current  and  magnetism  is  merely  empirical,  given  by 
the  hysteresis  cycle,  this  cannot  be  done  analytically,  but  only 
by  the  calculation  or  construction  of  the  instantaneous  values 
of  the  curves. 

249.  For  the  hysteresis  cycle  in  Fig.  180,  and  for  a  system  of 
transformers  connected  in  Y,  with  open  secondary  circuit,  into 
a  three-phase  system  with  a  sine  wave  of  e.m.f.  between  the 
lines,  the  curves  of  exciting  current,  magnetic  flux  and  voltage 
per  transformer,  or  between  lines  and  neutral,  are  constructed  in 
Fig.  184. 

i  is  the  exciting  current  of  the  transformer,  and  contains  all 
the  harmonics,  except  the  third  and  its  multiples.  It  is  given 
by  the  equation: 

i  =  8.28  sin  (0  +  30.8°)  -  0.71  sin  (5  <f>  -  17.2°)  +  .  ' .    . 


366 


AL  TERN  A  TING-C  URRENT  PHENOMENA 


B  is  the  magnetic  flux  density  in  the  transformer.  It  contains 
only  the  third  harmonic  and  its  multiples,  but  no  other  harmonics, 
and  is  given  by  the  equation: 

B  =  10.0  sin  0  +  1.38  sin  (3  <f>  -  9.2°)  +  0.045  sin  9  0  +  .    .   . 

e  is  the  potential  difference  of  the  transformer  terminals,  or 
voltage  between  the  three-phase  lines  and  the  transformer  neu- 
tral. It  contains  the  third  harmonic  and  its  multiples,  but  no 
other  harmonics,  and  is  given  by  the  equation: 

e  =  10.0  cos  <£  +  4.14  cos  (3  <£  -  9.2°)  -j- 0.405  cos  9  0  +  . 


t 


\ 


FIG.  184. 

The  effective  value  of  the  voltage  is  0.625  e,  and  the  maximum 
value  is  1.175  E,  where  E  =  supply  voltage  or  A  voltage. 
While  with  a  sine  wave  the  effective  value  would  be 


and  the  maximum  value 


==  0.577  E, 


=  0.815 


V3 


that  is,  by  the  suppression  of  the  third  harmoniS  of  exciting  cur- 
rent in  the  three-phase  system,  the  effective  value  of  the  voltage 
per  transformer,  or  voltage  between  three-phase  lines  and  neutral 
(or  ground,  if  the  neutral  is  grounded)  has  been  increased  by 


DISTORTION  OF  WA  VE-SHAPE  AND  ITS  CA  USES   367 

8.5  per  cent.,  the  maximum  value  by  44.6  per  cent.,  and  the 
voltage  wave  has  become  very  peaked,  by  a  pronounced  third 
harmonic  of  an  effective  value  of  0.24  E — that  is,  38.5  per  cent, 
of  the  effective  value  of  the  total  wave. 

The  very  high  peak  of  e.m.f.  produced  by  this  wave-shape 
distortion  is  liable  to  be  dangerous  in  high-potential,  three- 
phase  systems  by  increasing  the  strain  on  the  insulation  between 
lines  and  ground,  and  leading  to  resonance  phenomena  with  the 
third  harmonic. 

The  maximum  value  of  the  distorted  wave  of  magnetism  is 
8.89,  while  with  a  sine  wave  it  would  be  10.0,  that  is,  the  maxi- 
mum of  the  wave  of  magnetism  has  been  reduced  by  11.1  per 
cent.,  and  the  core  loss  of  the  transformer  so  by  about  17  per 
cent. 

250.  Assuming  now  that  in  such  transformers,  connected 
with  their  primaries  in  Y  into  a  three-phase  circuit,  the  seconda- 
ries are  connected  in  A.  The  third  harmonics  of  e.m.f.,  generated 
in  the  three  transformer  secondaries,  then  are  in  series  in  short- 
circuit,  thus  produce  a  local  current  in  the  secondary  transformer 
triangle.  This  current  is  of  triple  frequency,  and  hence  supplies 
the  third  harmonic  of  exciting  .current,  which  was  suppressed  in 
the  primary,  and  thereby  eliminates  the  third  harmonic  of  mag- 
netism and  of  e.m.f.,  which  results  from  the  suppression  of  the 
third  harmonic  of  exciting  current,  and  so  limits  itself.  That  is, 
connecting  the  transformer  secondaries  in  A,  the  wave-shape  dis- 
tortion disappears,  and  voltage  and  magnetism  are  again  sine 
waves,  and  the  exciting  current  is  that  corresponding  to  a  sine 
wave  of  magnetism,  except  that  it  is  divided  between  primary 
and  secondary;  the  third  harmonic  of  the  exciting  current  does 
not  exist  in  the  primary,  but  is  produced  by  induction  in  the 
secondary  circuit.  Obviously,  in  this  case  the  magnetic  flux 
and  the  voltage  are  not  perfect  sine  waves,  but  contain  a  slight 
third  harmonic,  which  produces  in  the  secondary  the  triple- 
frequency  exciting  current. 

If  the  primary  neutral  of  the  transformers  is  connected  to  a 
fourth  wire,  in  a  four-wire,  three-phase  system  or  three-phase 
system  with  grounded  neutral,  and  this  fourth  wire  leads  back 
to  the  generator  neutral,  or  a  neutral  of  a  transformer  in  which 
the  triple-frequency  current  can  exist,  that  is,  in  which  the 
secondary  is  connected  in  A,  the  wave-shape  distortion  also 
disappears. 


368         ALTERNATING-CURRENT  PHENOMENA 

It  follows  herefrom  that  in  the  three-phase  system  attention 
must  be  paid  to  provide  a  path  for  the  third  harmonic  of  the 
transformer  exciting  current,  either  directly  or  inductively, 
otherwise  a  serious  distortion  of  the  e.m.f.  wave  of  the  trans- 
formers occurs.  (See  "  Theoretical  Elements  of  Electrical 
Engineering,"  Chapter  X.) 


CHAPTER  XXVI 
EFFECTS  OF  HIGHER  HARMONICS 

251.  To  elucidate  the  variation  in  the  shape  of  alternating 
waves  caused  by  various  harmonics,  in  Figs.  185  and  186  are 
shown  the  wave-forms  produced  by  the  superposition  of  the 


FIG.  185. 

triple  and  the  quintuple  harmonic  upon  the  fundamental  sine 
wave. 

In  Fig.  185  is  shown  the  fundamental  sine  wave  and  the  com- 
plex waves  produced  by  the  superposition  of  a  triple  harmonic 
of  30  per  cent,  the  amplitude  of  the  fundamental,  under  the  rela- 
24  369 


370 


ALTERNATING-CURRENT  PHENOMENA 


tive  phase  displacments  of  0°,  45°,  90°,  135°,  and  180°,  repre- 
sented by  the  equations: 

sin  j8 

sin  ]8  -  0.3  sin  3  0 

sin  0  -  0.3  sin  (3  0  -  45°) 

sin  0  —  0.3  sin  (3  0  -  90°) 

sin  0  -  0.3  sin  (3  0  -  135°) 

sin  0  -  0.3  sin  (30-  180°). 


Dfstortion  of  Wave  Shape 
by  Quintuple  Harmonic 
Sin./?-2siru(5/?-,6)  > 


FIG.  186. 

As  seen,  the  effect  of  the  triple  harmonic  is,  in  the  first  figure, 
to  flatten  the  zero  values  and  point  the  maximum  values  of  the 
wave,  giving  what  is  called  a  peaked  wave.  With  increasing 
phase  displacement  of  the  triple  harmonic,  the  flat  zero  rises  and 
gradually  changes  to  a  second  peak,  giving  ultimately  a  flat-top 
or  even  double-peaked  wave  with  sharp  zero.  The  intermediate 
positions  represent  what  is  called  a  saw-tooth  wave. 

In  Fig.  186  are  shown  the  fundamental  sine  wave  and  the 


EFFECTS  OF  HIGHER  HARMONICS 


371 


complex  waves  produced  by  superposition  of  a  quintuple  har- 
monic of  20  per  cent,  the  amplitude  of  the  fundamental,  under  the 
relative  phase  displacement  of  0°,  45°,  90°,  135°,  180°,  represented 
by  the  equations: 

sin  (3 

sin  0  —  0.2  sin  5  0 

sin  0  -  0.2  sin  (50-  45°) 

sin  0  -  0.2  sin  (50-  90°) 

sin  0  -  0.2  sin  (5  0  -  135°) 

sin  0  -  0.2  sin  (50  -  180°). 


FIG.  187. — Some  characteristic  wave-shapes. 

The  quintuple  harmonic  causes  a  flat-topped  or  even  double- 
peaked  wave  with  flat  zero.  With  increasing  phase  displacement 
the  wave  becomes  of  the  type  called  saw-tooth  wave  also.  The 
flat  zero  rises  and  becomes  a  third  peak,  while  of  the  two  former 


372         ALTERNATING-CURRENT  PHENOMENA 

peaks,  one  rises,  the  other  decreases,  and  the  wave  gradually 
changes  to  a  triple-peaked  wave  with  one  main  peak,  and  a  sharp 
zero. 

As  seen,  with  the  triple  harmonic,  flat  top  or  double  peak 
coincides  with  sharp  zero,  while  the  quintuple  harmonic  flat  top 
or  double  peak  coincides  with  flat  zero. 

Sharp  peak  coincides  with  flat  zero  in  the  triple,  with  sharp 
zero  in  the  quintuple  harmonic.  With  the  triple  harmonic,  the 
saw-tooth  shape  appearing  in  case  of  a  phase  difference  between 
fundamental  and  harmonic  is  single,  while  with  the  quintuple 
harmonic  it  is  double. 

Thus  in  general,  from  simple  inspection  of  the  wave-shape, 
the  existence  of  these  first  harmonics  can  be  discovered.  Some 
characteristic  shapes  are  shown  in  Fig.  187. 

Flat  top  with  flat  zero, 

sin  0  -  0.15  sin  3  0  -  0.10  sin  5  0. 

Flat  top  with  sharp  zero, 

sin  0  -  0.225  sin  (30-  180°)  -  0.05  sin  (5  ft  -  180°). 

Double  peak,  with  sharp  zero, 

sin  0-0.15  sin  (3  0  -  180°)  -  0.10  sin  5  0. 

Sharp  peak  with  sharp  zero, 

sin  0  -  0.15  sin  3  0  -  0.10  sin  (5  0  -  180°). 

For  further  discussion  of  wave-shape  distortion  by  harmonics 
see  "  Engineering  Mathematics." 

252.  Since  the  distortion  of  the  wave-shape  consists  in  the 
superposition  of  higher  harmonics,  that  is,  waves  of  higher  fre- 
quency, the  phenomena  taking  place  in  a  circuit  supplied  by  such 
a  wave  will  be  the  combined  effect  of  the  different  waves. 

Thus  in  a  non-inductive  circuit  the  current  and  the  potential 
difference  across  the  different  parts  of  the  circuit  are  of  the  same 
shape  as  the  impressed  e.m.f.  If  inductive  reactance  is  inserted 
in  series  with  a  non-inductive  circuit,  the  self-inductive  reactance 
consumes  more  e.m.f.  of  the  higher  harmonics,  since  the  reactance 
is  proportional  to  the  frequency,  and  thus  the  current  and  the 
e.m.f.  in  the  non-inductive  part  of  the  circuit  show  the  higher 
harmonics  in  a  reduced  amplitude.  That  is,  self-inductive  react- 
ance in  series  with  a  non-inductive  circuit  reduces  the  higher 
harmonics  or  smooths  out  the  wave  to  a  closer  resemblance  to 
sine-shape.  Inversely,  capacity  in  series  to  a  non-inductive 
circuit  consumes  less  e.m.f.  at  higher  than  at  lower  frequency, 
and  thus  makes  the  higher  harmonics  of  current  and  of  potential 


EFFECTS  OF  HIGHER  HARMONICS  373 

difference  in  the  non-inductive  part  of  the  circuit  more   pro- 
nounced— intensifies  the  harmonics. 

Self-induction  and  capacity  in  series  may  cause  an  increase  of 
voltage  due  to  complete  or  partial  resonance  with  higher  har- 
monics, and  a  discrepancy  between  volt-amperes  and  watts, 
without  corresponding  phase  displacement,  as  will  be  shown 
hereafter. 

253.  In  long-distance  transmission  over  lines  of  noticeable 
inductive  and  condensive  reactance,  rise  of  voltage  due  to  reso- 
nance may  occur  with  higher  harmonics,  as  waves  of  higher  fre- 
quency, while  the  fundamental  wave  is  usually  of  too  low  a 
frequency  to  cause  resonance. 

An  approximate  estimate  of  the  possible  rise  by  resonance  with 
various  harmonics  can  be  obtained  by  the  investigation  of  a 
numerical  example.  Let  in  a  long-distance  line,  fed  by  step-up 
transformers  at  60  cycles, 

The  resistance  drop  in  the  transformers  at  full-load  =  1  per  cent. 
The  reactance  voltage  in  the  transformers  at  full-load  =  5  per 

cent,  with  the  fundamental  wave. 

The  resistance  drop  in  the  line  at  full-load  =  10  per  cent. 
The  reactance  voltage  in  the  line  at  full-load  =  20  per  cent,  with 

the  fundamental  wave. 

The  capacity  or  charging  current  of  the  line  =  20  per  cent. -of  the 
full-load  current,  /,  at  the  frequency  of  the  fundamental. 

The  line  capacity  may  approximately  be  represented  by  a 
condenser  shunted  across  the  middle  of  the  line.  The  e.m.f.  at 
the  generator  terminals,  E,  is  assumed  as  maintained  constant. 

The  e.m.f.  consumed  by  the  resistance  of  the  circuit  from 
generator  terminals  to  condenser  is 

IT  =  0.06  E, 
or, 

r  =  0.06  j- 

The  reactance  e.m.f.  between  generator  terminals  and  con- 
denser is,  for  the  fundamental  frequency, 

Ix  =  0.15#, 

or, 

x  =  0.15 - 


374         ALTERNATING-CURRENT  PHENOMENA 

thus  the  reactance  corresponding  to  the  frequency  (2  A;  —  I)/  of 
the  higher  harmonic  is 

x  (2k-  1)  =  0.15  (2  k-  l)j- 
The  capacity  current  at  fundamental  frequency  is, 

i  =  0.27; 
hence,  at  the  frequency  (2k—  I)/, 

i  =  0.2  (2  k  -  1)  e'  ^ 
if 

e'  =  e.m.f.  of  the  (2  k  —  l)th  harmonic  at  the  condenser, 
e    =  e.m.f.  of  the  (2  k  —  l)th  harmonic  at  the  generator  terminals. 
The  e.m.f.  at  the  condenser  is 


e'  =  v^^72r2  +  ix(2k  -  1); 
hence,  substituted, 

e'  1 


Vl  -  0.059856(2  k  -  I)2  +  0.0009  (2  A;  -  I)4 

the  rise  of  voltage  by  inductive  and  condensive  reactance. 
Substituting, 

k  =        1  2  3  4  5  6 

or,      2k  -  1  =        1  3  5  7  9          11 

and  a  =  1.03       1.36      3.76      2.18      0.70      0.38 

That  is,  the  fundamental  will  be  increased  at  open  circuit  by 
3  per  cent.,  the  triple  harmonic  by  36  per  cent.,  the  quintuple 
harmonic  by  276  per  cent.,  the  septuple  harmonic  by  118  per 
cent.,  while  the  still  higher  harmonics  are  reduced. 

The  maximum  possible  rise  will  take  place  for 


that  is,  at  a  frequency  /  =  346,  and  a  =  14.4. 

That  is,  complete  resonance  will  appear  at  a  frequency  between 
quintuple  and  septuple  harmonic,  and  would  raise  the  voltage  at 
this  particular  frequency  14.4-fold. 

If  the  voltage  shall  not  exceed  the  impressed  voltage  by  more 
than  100  per  cent.,  even  at  coincidence  of  the  maximum  of  the 
harmonic  with  the  maximum  of  the  fundamental, 


EFFECTS  OF  HIGHER  HARMONICS  375 

the  triple  harmonic    must    be  less   than    70   per  cent,   of   the 

fundamental, 
the  quintuple  harmonic  must  be  less  than  26.5  per  cent,  of  the 

fundamental, 
the  septuple  harmonic  must  be  less  than  46  per  cent,  of  the 

fundamental. 

The  voltage  will  not  exceed  twice  the  normal,  even  at  a  fre- 
quency of  complete  resonance  with  the  higher  harmonic,  if  none 
of  the  higher  harmonics  amounts  to  more  than  7  per  cent,  of  the 
fundamental.  Herefrom  it  follows  that  the  danger  of  resonance 
in  high-potential  lines  is  frequently  overestimated,  since  the 
conditions  assumed  in  this  example  are  rather  more  severe  than 
found  in  lines  of  moderate  length,  the  capacity  current  of  such 
line  very  seldom  reaching  20  per  cent,  of  the  main  current. 

254.  The  power  developed  by  a  complex  harmonic  wave  in  a 
non-inductive  circuit  is  the  sum  of  the  powers  of  the  individual 
harmonics.     Thus  if  upon   a   sine   wave   of  alternating   e.m.f. 
higher  harmonic  waves  are  superposed,  the  effective  e.m.f.  and 
the  power  produced  by  this  wave  in  a  given  circuit  or  with  a  given 
effective  current  are  increased.     In  consequence  hereof  alterna- 
tors and  synchronous  motors  of  iron-clad  unitooth  construction 
—  that    is,    machines    giving    waves    with    pronounced    higher 
harmonics  —  may  give  with  the  same  number  of  turns,  on  the 
armature,  and  the  same  magnetic  flux  per  field-pole  at  the  same 
frequency,  a  higher  output  than  machines  built  to  produce  sine 
waves. 

255.  This  explains  an  apparent  paradox: 

If  in  the  three-phase  star-connected  generator  with  the  mag- 
netic field  constructed  as  shown  diagrammatically  in  Fig.  188 
the  magnetic  flux  per  pole  =  3>,  the  number  of  turns  in  series 
per  circuit  =  n,  the  frequency  =  /,  the  e.m.f.  between  any 
two  collector  rings  is 

E  =  V2  TT/  2  n$  10-8, 

since    2  n    armature    turns    simultaneously  interlink    with   the 
magnetic  flux,  $. 

The  e.m.f.  per  armature  circuit  is 


hence  the  e.m.f.  between  collector  rings,  as  resultant  of  two 
e.m.fs.,  e,  displaced  by  60°  from  each  other,  is 

E  =  e-v/3 


376         ALTERNATING-CURRENT  PHENOMENA 

while  the  same  e.m.f.  was  found  from  the  number  of  turns,  the 
magnetic  flux,  and  the  frequency  by  direct  calculation  to  be 
equal  to  2  e]  that  is,  the  two  values  found  for  the  same  e.m.f. 
have  the  proportion  \/3'2  =  1 : 1.154. 

This  discrepancy  is  due  to  the  existence  of  more  pronounced 
higher  harmonics  in  the  wave  e  than  in  the  wave  E  =  e  X  \/3> 
which  have  been  neglected  in  the  formula 

e  =  \/2irfn3>W-&. 

Hence  it  follows  that,  while  the  e.m.f.  between  two  collector 
rings  in  the  machine  shown  diagrammatically  in  Fig.  188  is  only 


FIG.  188. — Three-phase  star-connected  alternator. 

e  X  \/3j  by  massing  the  same  number  of  turns  in  one  slot 
instead  of  in  two  slots,  we  get  the  e.m.f.  2e,  or  15.4  per  cent, 
higher  e.m.f.,  that  is,  larger  output. 

It  follows  herefrom  that  the  distorted  e.m.f.  wave  of  a  unitooth 
alternator  is  produced  by  lesser  magnetic  flux  per  pole — that  is, 
in  general,  at  a  lesser  hysteretic  loss  in  the  armature  or  at  higher 


EFFECTS  OF  HIGHER  HARMONICS  377 

efficiency — than  the  same  effective  e.m.f.  would  be  produced 
with  the  same  number  of  armature  turns  if  the  magnetic  dispo- 
sition were  such  as  to  produce  a  sine  wave. 

256.  Inversely,  if  such  a  distorted  wave  of  e.m.f.  is  impressed 
upon  a  magnetic  circuit,  as,  for  instance,  a  transformer,  the  wave 
of  magnetism  in  the  primary  will  repeat  in  shape  the  wave  of 
magnetism  interlinked  with  the  armature  coils  of  the  alternator, 
and    consequently  with  a  lesser  maximum   magnetic   flux  the 
same  effective  counter  e.m.f.  will  be  produced,  that  is,  the  same 
power  converted  in  the  transformer.     Since  the  hysteretic  loss 
in  the  transformer  depends  upon  the  maximum  value  of  mag- 
netism, it  follows  that  the  hysteretic  loss  in  a  transformer  is  less 
with  a  distorted  wave  of  a  unitooth  alternator  than  with  a  sine 
wave. 

257.  From  another  side  the  same  problem  can  be  approached : 
If  upon  a  transformer  a  sine  wave  of  e.m.f.  is  impressed,  the 
wave  of  magnetism  will  be  a  sine  wave  also.     If  now  upon  the 
sine  wave  of  e.m.f.  higher  harmonics,  as  sine  waves  of  triple, 
quintuple,  etc.,  frequency  are  superposed  in  such  a  way  that  the 
corresponding  higher  harmonic  sine  waves  of  magnetism  do  not 
increase  the  maximum  value  of  magnetism,  or  even  lower  it  by  a 
coincidence  of  their  negative  maxima  with  the  positive  maximum 
of  the  fundamental,  in  this  case  all  the  power  represented  by 
these  higher  harmonics  of  e.m.f.  will  be  transformed  without  an 
increase  of  the  hysteretic  loss,  or  even  with  a  decreased  hysteretic 
loss. 

Obviously,  if  the  maximum  of  the  higher  harmonic  wave  of 
magnetism  coincides  with  the  maximum  of  the  fundamental,  and 
thereby  makes  the  wave  of  magnetism  more  pointed,  the  hyster- 
etic loss  will  be  increased  more  than  in  proportion  to  the  in- 
creased power  transformed,  i.e.,  the  efficiency  of  the  transformer 
will  be  lowered. 

That  is,  some  distorted  waves  of  e.m.f.  are  transformed  at  a 
lesser,  some  at  a  larger,  hysteretic  loss  than  the  sine  wave,  if  the 
same  effective  e.m.f.  is  impressed  upon  the  transformer. 

The  unitooth  alternator  wave  and  the  first  wave  in  Fig.  226 
belong  to  the  former  class;  the  waves  derived  from  continuous- 
current  machines,  tapped  at  two  equidistant  points  of  the 
armature,  frequently,  to  the  latter  class. 

258.  Regarding  the  loss  of  energy  by  Foucault  or  eddy  currents, 
this  loss  is  not  affected  by  distortion  of  wave-shape,  since  the 


378         ALTERNATING-CURRENT  PHENOMENA 

e.m.f.  of  eddy  currents,  like  the  generated  e.m.f.,  is  proportional 
to  the  secondary  e.m.f. ;  and  thus  at  constant  impressed  primary 
e.m.f.  the  power  consumed  by  eddy  currents  bears  a  constant 
relation  to  the  output  of  the  secondary  circuit,  as  obvious,  since 
the  division  of  power  between  the  two  secondary  circuits — the 
eddy-current  circuit  and  the  useful  or  consumer  circuit — is 
unaffected  by  wave-shape  or  intensity  of  magnetism. 

In  high-potential  lines,  distorted  waves  whose  maxima  are 
very  high  above  the  effective  values,  as  peaked  waves,  are 
objectionable  by  increasing  the  strain  on  the  insulation.  The 
striking-distance  of  an  alternating  voltage  depends  upon  the 
maximum  value,  except  at  extremely  high  frequencies,  such  as 
oscillating  discharges.  In  the  latter,  the  very  short  duration  of 
the  voltage  peak  may  reduce  the  disruptive  strength,  as  dielectric 
disruption  requires  energy,  that  is,  not  only  voltage,  but  time 
also. 


CHAPTER  XXVII 

SYMBOLIC  REPRESENTATION  OF  GENERAL 
ALTERNATING  WAVES 

259.  The  vector  representation, 

A  =  a1  +  ja11  =  a  (cos  6  +  j  sin  6) 
of  the  alternating  wave, 

A  =  aQ  cos  (0  —  6) 

applies  to  the  sine  wave  only. 

The  general  alternating  wave,  however,  contains  an  infinite 
series  of  terms,  of  odd  frequencies, 

A  =  Ai  cos  (  0-  0i)  +  A3  cos  (30  —  03)  +  A5  cos  (5  0  -  05)  + 
thus  cannot  be  directly  represented  by  one  complex  vector 
quantity. 

The  replacement  of  the  general  wave  by  its  equivalent  sine 
wave,  as  before  discussed,  that  is,  a  sine  wave  of  equal  effective 
intensity  and  equal  power,  while  sufficiently  accurate  in  many 
cases,  completely  fails  in  other  cases,  especially  in  circuits  con- 
taining capacity,  or  in  circuits  containing  periodically  (and  in 
synchronism  with  the  wave)  varying  resistance  or  reactance  (as 
alternating  arcs,  reaction  machines,  synchronous  induction 
motors,  oversaturated  magnetic  circuits,  etc.). 

Since,  however,  the  individual  harmonics  of  the  general  alter- 
nating wave  are  independent  of  each  other,  that  is,  all  products 
of  different  harmonics  vanish,  each  term  can  be  represented  by  a 
complex  symbol,  and  the  equations  of  the  general  wave  then  are 
the  resultants  of  those  of  the  individual  harmonics. 

This  can  be  represented  symbolically  by  combining  in  one 
formula  symbolic  representations  of  different  frequencies,  thus, 


1  The  index  2n  —  1  in  the  S  sign  denotes  that  only  the  odd  values  of 
n  are  considered.  If  the  wave  contained  even  harmonics,  the  even  values 
of  n  would  also  be  considered,  and  the  index  in  the  S  sign  would  be  n. 

379 


380         ALTERNATING-CURRENT  PHENOMENA 
where 


and  the  index  of  the  jn  merely  denotes  that  the  j's  of  differ- 
ent indices,  n,  while  algebraically  identical,  physically  represent 
different  frequencies,  and  thus  cannot  be  combined. 
The  general  wave  of  e.m.f.  is  thus  represented  by 

E  =  &•-!<«.*  +**.")« 

i 

the  general  wave  of  current  by 

I    =    S2»-1(»V   +jnin11). 
If 

Zi  =  r  +  j  (xm  +  z0  +  xc) 

is  the  impedance  of  the  fundamental  harmonic,  where 

xm  is  that  part  of  the  reactance  which  is  proportional  to  the 
frequency  (inductance,  etc.). 

XQ  is  that  part  of  the  reactance  which  is  independent  of  the 
frequency  (mutual  inductance,  synchronous  motion,  etc.). 

xc  is  that  part  of  the  reactance  which  is  inversely  propor- 
tional to  the  frequency  (capacity,  etc.). 

The  impedance  for  the  nth  harmonic  is 


Z  =  r  +  jn  (nx 


This  term  can  be  considered  as  the  general  symbolic  expression 
of  the  impedance  of  a  circuit  of  general  wave-shape. 

Ohm's  law,  in  symbolic  expression,  assumes  for  the  general 
alternating  wave  the  form 


=      or, 


E  =  IZor,  S2»-i  (e,1  +j»c.»)  =  22  —  1  \r  +  j.  (nxm  +  %  +  ^ 

j  j  \  74 


„       E  .  I  xc\       en+jnen11 

OTZn  =  r+ 


The  symbols  of  multiplication  and  division  of  the  terms,  E,  I,  Z, 


GENERAL  ALTERNATING  WAVES      381 

thus  represent,  not  algebraic  operation,  but  multiplication  and 
division  of  corresponding  terms  of  E,  I,  Z,  that  is,  terms  of  the 
same  index,  n,  or,  in  algebraic  multiplication  and  division  of  the 
series,  E,  I,  all  compound  terms,  that  is,  terms  containing  two 
different  n's,  vanish. 

260.  The  effective  value  of  the  general  wave, 
a  =  Ai  cos  (<£  —  00  +  A3  cos  (3  0  —  03)  +  Ab  cos  (5  0  —  02)  +.  . 
is  the  square  root  of  the  sum  of  mean  squares  of  individual  har- 
monics, 

A  = 


Since,  as  discussed  above,  the  compound  terms  of  two  different 
indices,  n,  vanish,  the  absolute  value  of  the  general  alternating 
wave, 


A  =  22«-: 
is  thus, 

A    =         /T.**_ t°»       ~t~  ^n 


which   offers   an   easy   means   of  reduction   from   symbolic   to 
absolute  values. 

Thus,  the  absolute  value  of  the  e.m.f., 


E  =  S2n-i(eni+jnenii), 
i 


IS 


the  absolute  value  of  the  current, 


is 


J22n-l(tr 


261.  The  double  frequency  power  (torque,  etc.)  equation  of 
the  general  alternating  wave  has  the  same  symbolic  expression 
as  with  the  sine  wave, 


382         ALTERNATING-CURRENT  PHENOMENA 

P  =  [El] 
=  P1  '+  jP* 


i 
where 


The  jn  enters  under  the  summation  sign  of  the  reactive  or 
"wattless  power,"  P',  so  that  the  wattless  powers  of  the  different 
harmonics  cannot  be  algebraically  added. 

Thus, 

The  total  "true  power"  of  a  general  alternating-current  circuit 
is  the  algebraic  sum  of  the  powers  of  the  individual  harmonics. 

The  total  "reactive  power"  of  a  general  alternating-current 
circuit  is  not  the  algebraic,  but  the  absolute  sum  of  the  wattless 
powers  of  the  individual  harmonics. 

Thus,  regarding  the  reactive  power  as  a  whole,  in  the  general 
alternating  circuit  no  distinction  can  be  made  between  lead  and 
lag,  since  some  harmonics  may  be  leading,  others  lagging. 

The  apparent  power,  or  total  volt-amperes,  of  the  circuit  is 


pa  =  El  = 

i  i 

The  power-factor  of  the  circuit  is, 

^n-i^HV  + 
P1  i 


The  term  " inductance  factor,"  however,  has  no  meaning  any 
more,  since  the  reactive  powers  of  the  different  harmonics  are  not 
directly  comparable. 

The  quantity 

qQ  =  Vl  ~p2 

,...._  .  .  reactive  power 

has  no  physical  S1gmficance,  and  is  not  total  apparent  power- 


GENERAL  ALTERNATING  WAVES  383 

The  term 


m    r     3       EI 

v«o      ,3n  „ 


where 

enllinl  -  enlin11 


consists  of  a  series  of  inductance  factors,  qn,  of  the  individual 
harmonics. 

CO 

As  a  rule,  if  q2  —  2)2n-igfn2? 

i 

p2  +  q2  <  1, 
for  the  general  alternating  wave,  that  is,  q  differs  from 


The  complex  quantity, 


Pa  "   El  El 


i 

takes  in  the  circuit  of  the  general  alternating  wave  the  same 
position  as  power-factor  and  inductance  factor  with  the  sine  wave. 

p 

V  =  p-  may  be  called  the  "circuit-factor." 

It  consists  of  a  real  term,  p,  the  power-factor,  and  a  series  of 
imaginary  terms,  jnqn,  the  inductance  factors  of  the  individual 
harmonics. 

The  absolute  value  of  the  circuit-factor, 


v 
as  a  rule,  is  <  1. 


384         ALTERNATING-CURRENT  PHENOMENA 

262.  Some   applications   of  this   symbolism   will   explain   its 
mechanism  and  its  usefulness  more  fully. 
First  Example. — Let  the  e.m.f., 

5 

E  =  Z2«-i(eni  +  j«eir), 

i 

be  impressed  upon  a  circuit  of  the  impedance, 

Z  =  r  +  jn  (nxm  -  J)  -  10+j.(lOn-  f ) 

that  is,  containing  resistance,  r,  inductive  reactance  xm  and  con- 

densive  reactance  xc  in  series. 

Let 

ei1  =  720  ei11  =  -  540 

es1  =  283  e3u  =  283 

651  =  -  104  ebn  =  -  138 

or, 

ei    =  900  tan  Bl  =  0.75 

63    =  400  tan  03  =  -  1.0 

65    =  173  tan  05  =  -  1.33 

It  is  thus  in  symbolic  expression, 

Zi  =  10  -  80  ji  0i  =  80.6 

Z3  =  10  z3  =  10.0 

Z5  =  10  +  32J5  25  =  33.5, 
and  e.m.f., 

E  =  (720  -  540  j,)  +  (283  +  283  J3)  +  (  -  104  -  138  J5), 

or,  absolute, 

E  =  1000, 
and  current, 

E  _  720  -  540  j\      283  + 283  J3        -  104  -  138  jb 
'  Z=     10-SOji  10  10  +  32J5 

=  (7.76  +  8.04  JO  +  (28.3  +  28.3  J3)  +  (  -  4.86  +  1.73  J5) 

or,  absolute, 

/  =  41.85, 

of  which  is  of  fundamental  frequency, 

/i  =  11.15 
of  triple  frequency, 

73  =  40 


GENERAL  ALTERNATING  WAVES  385 

of  quintuple  frequency, 

75  =  5.17. 

The  total  apparent  power  of  the  circuit  is 

pa  =  El  =  41,850. 
The  true  power  of  the  circuit  is, 

pi  =  [El}1  =  1240  +  16,000  +  270, 
=  17,510, 

the  reactive  power, 

jP>'  =  j[EIV  =  -  10,000  ji  +  850  J5; 
thus,  the  total  power, 

P  =  17,510  -  10,000  ji  +  850  j5. 

That  is,  the  reactive  power  of  the  first  harmonic  is  leading, 
that  of  the  third  harmonic  zero,  and  that  of  the  fifth  harmonic 
lagging. 

17,510  =  Pr,  as  obvious. 

The  circuit-factor  is, 

V       t       \&1 
~  Pa  ~  El 

=  0.418  -  0.239  ji  4-  0.0203  j5, 
or,  absolute,  _  _ 

v  =  V0.4182  +  0.2392  +  0.02032. 
=  0.482. 

The  power-factor  is 

p  =  0.418. 

The  inductance  factor  of  the  first  harmonic  is  q\  =  —  0.239, 
that  of  the  third  harmonic  <?3  =  0,  and  of  the  fifth  harmonic 
g5  =  +  0.0203. 

Considering  the  waves  as  replaced  by  their  equivalent  sine 
waves,  from  the  sine  wave  formula, 

p2  +  9o2  -  1, 
the  inductance  factor  would  be, 

q0  =  0.914, 
and  the  phase  angle, 


25 


386         ALTERNATING-CURRENT  PHENOMENA 

giving  apparently  a  very  great  phase  displacement,  while  in 
reality,  of  the  41.85  amp.  total  current,  40  amp.  (the  current  of 
the  third  harmonic)  are  in  phase  with  their  e.m.f. 

We  thus  have  here  a  case  of  a  circuit  with  complex  harmonic 
waves  which  cannot  be  represented  by  their  equivalent  sine 
waves.  The  relative  magnitudes  of  the  different  harmonics  in 
the  wave  of  current  and  of  e.m.f.  differ  essentially,  and  the  circuit 
has  simultaneously  a  very  low  power-factor  and  a  very  low 
inductance  factor;  that  is,  a  low  power-factor  exists  without 
corresponding  phase  displacement,  the  circuit-factor  being  less 
than  one-half. 

Such  circuits,  for  instance,  are  those  including  alternating 
arcs,  reaction  machines,  synchronous  induction  motors,  react- 
ances with  over-saturated  magnetic  circuit,  high  potential  lines 
in  which  the  maximum  difference  of  potential  exceeds  the  corona 
voltage,  polarization  cells  and  in  general  electrolytic  conductors 
above  the  dissociation  voltage  of  the  electrolyte,  etc.  Such  cir- 
cuits cannot  correctly,  and  in  many  cases  not  even  approximately, 
be  treated  by  the  theory  of  the  equivalent  sine  waves,  but  re- 
quire the  symbolism  of  the  complex  harmonic  wave. 

263.  Second  Example. — A  condenser  of  capacity,  Co  =  20  mf .  is 
connected  into  the  circuit  of  a  60-cycle  alternator  giving  a  wave 
of  the  form, 

e  =  E(cos  0  -  0.10  cos  3  </>  -  0.08  cos  5  </>  +  0.06  cos  7  0), 
or,  in  symbolic  expression, 

E  =  e(li  -  0.103  -  0.085  +  0.067). 
The  synchronous  impedance  of  the  alternator  is 
ZQ  =  r0  +  jnnxQ  =  0.3  +  5  njn. 

What  is  the  apparent  capacity,  C,  of  the  condenser  (as  calcu- 
lated from  its  terminal  volts  and  amperes)  when  connected 
directly  with  the  alternator  terminals,  and  when  connected 
thereto  through  various  amounts  of  resistance  and  inductive 
reactance? 

The  condensive  reactance  of  the  condenser  is 

106 

*<  -  2tfCl  -  132  °hms' 
or,  in  symbolic  expression, 

•  5f  132  . 

3nn  -  n  Jn' 


GENERAL  ALTERNATING  WAVES  387 

Let  Zi  =  r  +  jnnx  =  impedance  inserted  in  series  with  the 
condenser. 

The  total  impedance  of  the  circuit  is  then 


Z  =  Zo  +  Zx-jn       =  (0.3  +  r)  +j 
The  current  in  the  circuit  is 

i  OJL 


_     r 

=  Z  =    e  10. 


_ 

(0.3  +  r)+j(x-  127)       (0.3  +  r)  +  j,  (3  x  -  29) 
0.08  0.06 


(0.3  +  r)  +  J5  (5z-1.4) 
and  the  e.m.f.  at  the  condenser  terminals, 

.  xj  f  132  J! 4.4  J3 

f  r)  +ji(x  -  127)     (0.3+r)+j3(3z-29) 

"  (03  +  r] T-l-  J5  (5s-  1.4)  +  (0.3  +  r)  +J7(7a:+  16.1) 
thus  the  apparent  condensive  reactance  of  the  condenser  is 

=  —  > 
and  the  apparent  capacity, 

c=  106 


27T/X1 

(a)  x  =  0:  Resistance,  r,  in  series  with  the  condenser.     Re- 
duced to  absolute  values  it  is 

1  0.01  0.0064  0.0036 


_  _  (0.3+r)  +  16129   (0.3+r)2+  841   (0.3+r)2+1.96   (0.3+r)2+259 
S  ~     16129          19.4     __  4.45          1.28 

(0.3+r)2+84l"i"(0.3+r)2+1.96"t"(0.3+r)2+259 


(6)  r  =  0:  Inductive  reactance,  x,  in  series  with  the  condenser. 
Reduced  to  absolute  values  it  is 

1     .     0.01         0.0064 


0.09+(a-127)2n0.09+(3a;-29)2n0.09+(5:c-1. 
"16129  19.4  4.45 


0.09+(a;-127)2"r0.09+(3a;-29)2+0.09+(5x-1.4) 


0.0036 


0.09+(7x+16.1)2. 

L28 

0.09+(7z+16.1)2 


388         ALTERNATING-CURRENT  PHENOMENA 


From  ~~i  are  derived  the  values  of  apparent  capacity, 

1O6 

C  = 


27T/Z! 

and  plotted  in  Fig.  189  for  values  of  r  and  x  respectively  varying 
from  0  to  22  ohms. 

As  seen,  with  neither  additional  resistance  nor  reactance  in 
series  to  the  condenser,  the  apparent  capacity  with  this  generator 
wave  is  84  mf.,  or  4.2  times  the  true  capacity,  and  gradually  de- 
creases with  increasing  series  resistance,  to  C  =  27  mf.  =  1.35 
times  the  true  capacity  at  r  =  13.2  ohms,  or  one-tenth  the  true 
capacity  reactance.  With  r  =  132  ohms,  or  with  an  additional 
resistance  equal  to  the  condensive  reactance,  C  =  20.2  mf .  or  only 


CAPACITY  C0=  20  mf    IN  CIRCUIT  OF  GENERATOR 
=E    (1-0.1 -0.08- 0.06)  OF  IMPEDANCE 
ftj,,  WITH  RESISTANCE  ,r  (I) 
OR  REACTANCE  X    (II)  IN  SERIES 


9   10  11  12  13  14  15  16  17  18  39  20 


FlG.  189. 

one  per  cent,  in  excess  of  the  true  capacity,  C0,  and  at  r  =  °° , 
C  =  20  mf.  or  the  true  capacity. 

With  reactance,  £,  but  no  additional  resistance,  r,  in  series,  the 
apparent  capacity,  C,  rises  from  4.2  times  the  true  capacity  at 
x  =  0,  to  a  maximum  of  5.03  times  the  true  capacity,  or  C  = 
100.6  mf.  at  x  =  0.28,  the  condition  of  resonance  of  the  fifth 
harmonic,  then  decreases  to  a  minimum  of  27  mf .,  or  35  per  cent, 
in  excess  of  the  true  capacity,  rises  again  to  60.2  mf.,  or  3.01 
times  the  true  capacity  at  x  =  9.67,  the  condition  of  resonance 
with  the  third  harmonic,  and  finally  decreases,  reaching  20  mf., 
or  the  true  capacity  at  x  —  132,  or  an  inductive  reactance  equal 
to  the  condensive  reactance. 


GENERAL  ALTERNATING  WAVES  389 

It  thus  follows  that  the  true  capacity  of  a  condenser  cannot 
even  approximately  be  determined  by  measuring  volts  and 
amperes  if  there  are  any  higher  harmonics  present  in  the  generator 
wave,  except  by  inserting  a  very  large  resistance  or  reactance 
in  series  to  the  condenser. 

264.  Third  Example. — An  alternating-current  generator  of  the 
wave, 

E0  =  2000  [li  +  0.123  -  0.235  -  0.137], 

and  of  synchronous  impedance, 

ZQ  =  0.3  +  5  njn, 
feeds  over  a  line  of  impedance, 

Zl  =  2  +  4  njn, 
a  synchronous  motor  of  the  wave, 

Ei  =  2250  [(cos  0  -  ji  sin  0)  +  0.24  (cos  3  0  -  j3  sin  3  0)], 
and  of  synchronous  impedance, 

Z2  =  0.3  +  6  njn. 
The  total  impedance  of  the  system  is  then, 

Z  =  Z0  +  Zi  +  Z2  =  2.6  +  15  njn, 
thus  the  current, 
.          .  .  7=^-=-^         ^     <    ~  ' 

2000-2250  cos  0+2250 j\ sin  0     240 -540 cos 3 0+540 j» sin 30 

2.6+15  ji  2.6-45J3 

460  260 

2.6  +  75  J6       2.6  +  105  J7 

where 

ai1    =  22.5  -  25.2  cos  0  +  146  sin  0, 

a3l    =  0.306  -  0.69  cos  3  0  +  11.9  sin  3  0, 

a-51    =  0.213, 

a?i    =  -  0.061, 

d11  =  130  -  146  cos  0  -  25.2  sin  0, 

o8»  =  5.3  -  11.9  cos  3  0  -  0.69  sin  3  0, 

a5n  =  _  6.12, 

a7n  =  _  2.48, 

or,  absolute, 


390         ALTERNATING-CURRENT  PHENOMENA 
first  harmonic, 


+  «ill2> 
third  harmonic, 


fifth  harmonic, 

ab  =  6.12, 
seventh  harmonic, 

a7  =  2.48, 


while  the  total  current  of  higher  harmonics  is 


/o  =  V  as2  +  a62  +  a72. 
The  true  input  of  the  synchronous  motor  is 

P1  =  [EJ]1 
=  (2250  a!1  cos  0+2250  ainsin  0)  +  (540  aj1  cos  3  0+540  a3usin  3  0 

=  Pi1  +  Ps1 

Pi1  =  2250  (a!1  cos  0  +  a!11  sin  0), 
is  the  power  of  the  fundamental  wave, 

Ps1  =  540  (as1  cos  3  0  +  a31>l  sin  3  0), 

the  power  of  the  third  harmonic. 

The  fifth  and  seventh  harmonics  do  not  give  any  power,  since 
they  are  not  contained  in  the  synchronous  motor  wave.  Sub- 
stituting now  different  numerical  values  for  0,  the  phase  angle 
between  generator  e.m.f.  and  synchronous  motor  counter  e.m.f., 
corresponding  values  of  the  currents,  /,  70,  and  the  powers,  P1, 
Pi1,  Ps1,  are  derived.  These  are  plotted  in  Fig.  190  with  the 
total  current,  I,  as  abscissas.  To  each  value  of  the  total  current, 
/,  correspond  two  values  of  the  total  power,  P1,  a  positive  value 
plotted  as  Curve  I — synchronous  motor — and  a  negative 
value  plotted  as  Curve  II — alternating-current  generator — . 
Curve  III  gives  the  total  current  of  higher  frequency,  /o,  Curve 
IV  the  difference  between  the  total  current  and  the  current  of 
fundamental  frequency,  7 — 7i,  in  percentage  of  the  total  current, 
I,  and  V  the  power  of  the  third  harmonic,  Ps1,  in  percentage  of 
the  total  power,  P1. 

Curves  III,  IV,  and  V  correspond  to  the  positive  or  synchron- 
ous motor  part  of  the  power  curve,  P1.  As  seen,  the  increase  of 


GENERAL  ALTERNATING  WAVES 


391 


current  due  to  the  higher  harmonics  is  small,  and  entirely  dis- 
appears at  about  180  amp.  The  power  of  the  third  harmonic 
is  positive,  that  is,  adds  to  the  power  of  the  synchronous  motor 


SYNCHRONOUS  MOTOR 
#,= 2250  (cos.  0-f  jisin.0)  + 
0.24 (cos.  30-hjs  sin.  3d 
OPERATED    FROM     GENERATOR 


^0=2000  ( 1-1-0.12-0.23-0.13 

OVER    TOTAL   IMPEDANCE 


FIG.  190. — Synchronous  motor. 

up  to  about  140  amp.  or  near  the  maximum  output  of  the  motor, 
and  then  becomes  negative. 

It  follows  herefrom  that  higher  harmonics  in  the  e.m.f.  waves 
of  generators  and  synchronous  motors  do  not  represent  a  mere 
waste  of  current,  but  may  contribute  more  or  less  to  the  output  of 


392         ALTERNATING-CURRENT  PHENOMENA 

the  motor.  Thus  at  75  amp.  total  current,  the  percentage  of 
increase  of  power  due  to  the  higher  harmonic  is  equal  to  the 
increase  of  current,  or  in  other  words  the  higher  harmonics 
of  current  do  work  with  the  same  efficiency  as  the  fundamental 
wave. 

265.  Fourth  Example.  —  In  a  small  three-phase  induction  motor, 
the  constants  per  delta  circuit  are 

primary  admittance  Y  =  0.002  -  0.03  j, 

self-inductive  impedance     Z0  =  Zi  =  0.6  +  2.4  j,  : 

and  a  sine  wave  of  e.m.f.,  eQ  =  110  volts,  is  impressed  upon  the 
motor. 

The  power  output,  P,  current  input,  Is,  and  power-factor,  p,  as 
function  of  the  slip,  s,  are  given  in  the  first  columns  of  the  follow- 
ing table,  calculated  in  the  manner  as  described  in  the  chapter  on 
Induction  Motors. 

To  improve  the  power-factor  of  the  motor  and  bring  it  to 
unity  at  an  output  of  500  watts,  a  condenser  capacity  is  required 
giving  4.28  amp.  leading  current  at  110  volts,  that  is,  neglecting 
the  power  loss  in  the  condenser,  capacity  susceptance 


In  this  case,  let  7,  =  current  input  into  the  motor  per  delta  cir- 
cuit at  slip  s,  as  given  in  the  following  table. 

The  total  current  supplied  by  the  circuit  with  a  sine  wave  of 
impressed  e.m.f.  is 

/'  =  J.  +  4.28J, 

power  current      .         . 

and  herefrom  the  power-factor  =    ,    .   *  --    —  ,  given  in  the 

total  current  ' 

second  columns  of  the  table. 

If  the  impressed  e.m.f.  is  not  a  sine  wave  but  a  wave  of  the 
shape, 

EQ  =  e0(li  +  0.123  -  0.238  -  0.1347), 

to  give  the  same  output,  the  fundamental  wave  must  be  the  same  : 
e0  =  110  volts,  when  assuming  the  higher  harmonics  in  the  motor 
as  wattless,  that  is, 

EQ    =  lid  +  13.23  -  25.35  -  14.77  =  eQ  +  EQ\ 


GENERAL  ALTERNATING  WAVES  393 

where  EQl  =  13.23  -  25.35  -  14.77 

=  component  of  impressed  e.m.f.  of  higher  frequency. 

The  effective  value  is 

EQ  =  114.5  volts. 

The  condenser  admittance  for  the  general  alternating  wave  is 
Yc  =  0.039  njn. 

Since  the  frequency  of  rotation  of  the  motor  is  small  com- 
pared with  the  frequency  of  the  higher  harmonics,  as  total 
impedance  of  the  motor  for  these  higher  harmonics  can  be 
assumed  the  stationary  impedance,  and  by  neglecting  the  resist- 
ance we  have 

Zl  =  njn(xQ  +  si)  =  4.8  njn 

The  exciting  admittance  of  the  motor,  fpr  these  higher  har- 
monics, is,  by  neglecting  the  conductance, 

yi  __     _bjn=     _  0.03  jn^ 
n  n 

and  the  higher  harmonics  of  counter  e.m.f., 


Thus  we  have, 

current  input  in  the  condenser, 

Ie  -  EQYC  =  +  4.28  ji  +  1.54  j,  -  4.93  J6  -  4.02  J7; 
high-frequency  component  of  motor-impedance  current, 

JV 

•^T  =  -  0.92  J3  +  1.06  J5  +  0.44  J7; 

it 

high-frequency  component  of  motor-exciting  current, 

EJY1 

^—  =  -  0.07  j3  +  0.08  j5  +  0.03  j7: 


thus,  total  high-frequency  component  of  motor  current, 


394         ALTERNATING-CURRENT  PHENOMENA 
and  total  current,  without  condenser, 

Io  =  Is  +  /o1  =  Is  -  0.99  J3  +  1-14  J6  +  0.47  J7, 
with  condenser, 
/  =  Ia  +  Jo1  -  Ic  =  L  +  4.28  J!  +  0.55  js  -  3.79  J5  +  3.55  J7; 

and  herefrom  the  power-factor. 

In  the  following  table  and  in  Fig.  191  are  given  the  values  of 
current  and  power-factor: 

I.  With  sine  wave  of  e.m.f.,  of  110  volts,  and  no  condenser. 

II.  With  sine  wave  of  e.m.f.,  of  110  volts,  and  with  condenser. 

III.  With  distorted  wave  of  e.m.f.,  of  114.5  volts,  and  no  condenser. 

IV.  With  distorted  wave  of  e.m.f.,  of  114.5  volts,  and  with  condenser. 

TABLE 
I.  II.  III.  IV. 


s  P                 I,                  I,         p         I  p         I0        p         I         p 

0.0  0        0.24+  3.10./    3.1     7.8     1.2  20.0    3.5    6.6     5.2    4.4 

0.01  160     1.73+  3. IQj    3.648.0    2.1  84.0    3.943.0    5.531.0 

0.02  320    3.32+  3.47 j    4.869.0    3.4  97.2     5.164.0     6.154.0 

0.035  500     5.16+  4.28j    6.7  77.0     5.2  100.0     6.9  72.5     7.2  68.0 

0.05  660     6.95+5.4.;      8.879.0    7.0  98.7     8.976.0.8.677.0 

0.07  810     8.77+  7.3  j    11.477.0    9.3  94.511.573.510.680.0 

0.10  885  10.1  +  9.85j  14.1  71.5  11.5  87.0  14.2  68.0  12.6  77.0 

0.13  900  10.45  +  11.45  j  15.5  67.5  12.7  82.015.664.513.773.0 

0.15  890  10. 75  +  12. 9j    16.8  64.0  13.8  78.0  16.9  61.0  14.7  70.0 

The  curves  II  and  IV  with  condenser  are  plotted  in  dotted  lines 
in  Fig.  191.  As  seen,  even  with  such  a  distorted  wave  the  current 
input  and  power-factor  of  the  motor  are  not  much  changed  if  no 
condenser  is  used.  When  using  a  condenser  in  shunt  to  the 
motor,  however,  with  such  a  wave  of  impressed  e.m.f.  the  increase 
of  the  total  current,  due  to  higher-frequency  currents  in  the  con- 
denser, is  greater  than  the  decrease,  due  to  the  compensation  of 
lagging  currents,  and  the  power-factor  is  actually  lowered  by  the 
condenser,  over  the  total  range  of  load  up  to  overload,  and  espe- 
cially at  light  load. 

Where  a  compensator  or  transformer  is  used  for  feeding  the 
condenser,  due  to  the  internal  self-inductance  of  the  compensa- 
tor, the  higher  harmonics  of  current  are  still  more  accentuated, 
that  is,  the  power-factor  still  more  lowered. 

In  the  preceding  the  energy  loss  in  the  condenser  and  compen- 
sator and  that  due  to  the  higher  harmonics  of  current  in  the  motor 


GENERAL  ALTERNATING  WAVES 


395 


has  been  neglected.  The  effect  of  this  energy  loss  is  a  slight 
decrease  of  efficiency  and  corresponding  increase  of  power-factor. 
The  power  produced  by  the  higher  harmonics  has  also  been 
neglected;  it  may  be  positive  or  negative,  according  to  the  index 


100  200  300 


400  600 


GO 


50 


40 


FIG.  191. 

of  the  harmonic,  and  the  winding  of  the  motor  primary.  Thus, 
for  instance,  the  effect  of  the  triple  harmonic  is  negative  in  the 
quarter-phase  motor,  zero  in  the  three-phase  motor,  etc.;  alto- 
gether, however,  the  effect  o£  these  harmonics  is  usually  small. 


SECTION  VII 
POLYPHASE  SYSTEMS 


CHAPTER  XXVIII 
GENERAL  POLYPHASE  SYSTEMS 

266.  A  polyphase  system  is  an  alternating-current  system  in 
which  several  e.m.fs.  of  the  same  frequency,  but  displaced  in 
phase  from  each  other,  produce  several  currents  of  equal  fre- 
quency, but  displaced  phases. 

Thus  any  polyphase  system  can  be  considered  as  consisting 
of  a  number  of  single  circuits,  or  branches  of  the  polyphase  sys- 
tem, which  may  be  more  or  less  interlinked  with  each  other. 

In  general  the  investigation  of  a  polyphase  system  is  carried 
out  by  treating  the  single-phase  branch  circuits  independently. 

Thus  all  the  discussions  on  generators,  synchronous  motors, 
induction  motors,  etc.,  in  the  preceding  chapters,  apply  to  single- 
phase  systems  as  well  as  polyphase  systems,  in  the  latter  case 
the  total  power  being  the  sum  of  the  powers  of  the  individual  or 
branch  circuits. 

If  the  polyphase  system  consists  of  n  equal  e.m.fs.  displaced 

from  each  other  by  -  of  a  period,  the  system  is  called  a  symmet- 
rical system,  otherwise  an  unsymmetrical  system. 

Thus  the  three-phase  system,  consisting  of  three  equal  e.m.fs. 
displaced  by  one-third  of  a  period,  is  a  symmetrical  system.  The 
quarter-phase  system,  consisting  of  two  equal  e.m.fs.  displaced 
by  90°,  or  one-quarter  of  a  period,  is  an  unsymmetrical  system. 

267.  The  power  in  a  single-phase  system  is  pulsating;  that  is, 
the  watt  curve  of  the  circuit  is  a  sine  wave  of  double  frequency, 
alternating  between  a  maximum  value  and  zero,  or  a  negative 
maximum  value.     In  a  polyphase  system  the  watt  curves  of  the 
different  branches  of  the  system  are  pulsating  also.     Their  sum, 
however,  or  the  total  power  of  the  system,  may  be  either  con- 

396 


GENERAL  POLYPHASE  SYSTEMS 


397 


stant  or  pulsating.     In  the  first  case,  the  system  is  called  a 
balanced  system,  in  the  latter  case  an  unbalanced  system. 

The  three-phase  system  and  the  quarter-phase  system,  with 
equal  load  on  the  different  branches,  are  balanced  systems;  with 
unequal  distribution  of  load  between  the  individual  branches 
both  systems  become  unbalanced  systems. 


FIG.  192. 

The  different  branches  of  a  polyphase  system  may  be  either 
independent  from  each  other,  that  is,  without  any  electrical  inter- 
connection, or  they  may  be  interlinked  with  each  other.  In  the 
first  case  the  polyphase  system  is  called  an  independent  system, 
in  the  latter  case  an  interlinked  system. 

The  three-phase  system  with  star-connected  or  ring-connected 
generator,  as  shown  diagrammatically  in  Figs.  192  and  193,  is  an 
interlinked  system. 


£"£ 


FIG.  193. 


The  four-phase  system  as  derived  by  connecting  four  equi- 
distant points  of  a  continuous-current  armature  with  four 
collector  rings,  as  shown  diagrammatically  in  Fig.  194,  is  an 
interlinked  system  also.  The  four-wire,  quarter-phase  system 
produced  by  a  generator  with  two  independent  armature  coils,  or 
by  two  single-phase  generators  rigidly  connected  with  each  other 
in  quadrature,  is  an  independent  system.  As  interlinked  system, 
it  is  shown  in  Fig.  195,  as  star-connected,  four-phase  system. 


398         ALTERNATING-CURRENT  PHENOMENA 

268.  Thus,  polyphase  systems  can  be  subdivided  into: 
Symmetrical  systems  and  unsymmetrical  systems. 
Balanced  systems  and  unbalanced  systems. 
Interlinked  systems  and  independent  systems. 
The  only  polyphase  systems  which  have  found  practical  appli- 
cation are: 


FIG.  194. 

The  three-phase  system,  consisting  of  three  e.m.fs.  displaced 
by  one-third  of  a  period,  is  used  exclusively  as  interlinked  system. 

The  quarter-phase  system,  consisting  of  two  e.m.fs.  in  quad- 
rature, and  used  with  four  wires,  or  with  three  wires,  which  may 
be  either  an  interlinked  system  or  an  independent  system. 

The  six-phase  system,  consisting  of  two  three-phase  systems 
in  opposition  to  each  other,  and  derived  by  transformation  from 


— E 


i 

0 

0 

^^~^                                                                 1 

(Jo  0  0  0  0  0  B  0 

,.,..,,c  +E                                                    ^ 

; 

^r 


FIG.  195. 


a  three-phase  system,  in  the  alternating  supply  circuit  of  large 
synchronous  converters. 

The  inverted  three-phase  system,  consisting  of  two  e.m.fs.  dis- 
placed from  each  other  by  60°,  and  derived  from  two  phases  of  a 
three-phase  system  by  transformation  with  two  transformers,  of 
which  the  secondary  of  one  is  reversed  with  regard  to  its  primary 
(thus  changing  the  phase  difference  from  120°  to  180°  -  120°  = 
60°)  finds  a  limited  application  in  low-tension  distribution. 


CHAPTER  XXIX 
SYMMETRICAL  POLYPHASE  SYSTEMS 

269.  If  all  the  e.m.fs.  of  a  polyphase  system  are  equal  in 
intensity  and  differ  from  each  other  by  the  same  angle  of  differ- 
ence of  phase,  the  system  is  called  a  symmetrical  polyphase 
system. 

Hence,  a  symmetrical  n-phase  system  is  a  system  of  n  e.m.fs. 

of  equal  intensity,  differing  from  each  other  in  phase  by  -  of  a 

period  : 

ei  =  E  sin 


i          2ir\ 
e2  =  E  sin  (0  ---  )  ; 

\          n  I 

e9  =  E  sin  h3  --  —)  ; 

•  •  \  7t>   / 


_  2(n-  1)  IT 
\  n 

The  next  e.m.f.  is,  again, 

ei  =  E  sin  (/3  —  2  TT)  =  #  sin  (8. 

In  the  vector  diagram  the  n  e.m.fs.  of  the  symmetrical  ft-phase 
system  are  represented  by  n  equal  vectors,  following  each  other 
under  equal  angles. 

Since  in  symbolic  writing  rotation  by  -  of  a  period,  or  angle 

2?r 
— ,  is  represented  by  multiplication  with 

lb 

27T   '        .     .       27T 

cos \-  j  sin  -  -  =  e, 

n  n 

the  e.m.fs.  of  the  symmetrical  polyphase  system  are 

E; 

E  (cos^I  +  ^in2^)  =E(. 

399 


400         ALTERNATING-CURRENT  PHENOMENA 

E   (  cos  ^  -1-  j  sin  ^)   =  Ee2; 

,-  /        2  (n  -  1)  TT   ,  2  (n  -  1)  ir\ 

#      cos  -          -^—  +  7  sin  —  ---  —  )  -  Ee1-1. 

•    \  n  n         ) 

The  next  e.m.f.  is  again, 

E(cos  2  TT  +  j  sin  2  TT)  =  Een  =  E. 
Hence,  it  is 

2*.         .      .        2»  „/- 

e  =  cos  —  +  j  sin  —  =  VI. 

/6  '6 

Or  in  other  words: 

In  a  symmetrical  n-phase  system  any  e.m.f.  of  the  system  is 
expressed  by 

JE; 
where 

«-  vT. 

270.  Substituting   now   for   n   different   values,   we   get  the 
different  symmetrical  polyphase  systems,  represented  by 


where 


n/-  27T  .      .       27T 

=  VI   =  cos  —  -  +  J  sin  —  • 

7t  Tt 


1.  n  =  1,    €  =  1,    c^  =  E, 
the  ordinary  single-phase  system. 

2.  n  =  2,    €  =  —  1,    e{E  =  E  and  —  #. 

Since  —  £7  is  the  return  of  E,  n  =  2  gives  again  the  single- 
phase  system. 

2w       .   .    27r       -  1  +  j  y/3 

3.  n  =  3,    e  =  cos  -j-  -f  j  sin  -     =  - 


2 

The  three  e.m.fs.  of  the  three-phase  system  are 


Consequently  the  three-phase  system  is  the  lowest  symmetrical 
polyphase  system. 


SYMMETRICAL  POLYPHASE  SYSTEMS          401 

2ir  2?r 

4.  n  =  4,  c  =  cos  -£-  +  j  sin  —  =  j,    £2  =  —  1,    e3  =  —  j. 

The  four  e.m.fs.  of  the  four-phase  system  are, 

^E  =  E,    JE,     -  E,    -jE. 
They  are  in  pairs  opposite  to  each  other, 

E  and  -  #;  jE  and  -  jE. 

Hence  can  be  produced  by  two  coils  in  quadrature  with  each 
other,  analogous  as  the  two-phase  system,  or  ordinary  alternating 
current  system,  can  be  produced  by  one  coil. 

Thus  the  symmetrical  quarter-phase  system  is  a  four-phase 
system. 

Higher  systems  than  the  quarter-phase  or  four-phase  system 
have  not  been  very  extensively  used,  and  are  thus  of  less  practical 
interest.  A  symmetrical  six-phase  system,  derived  by  trans- 
formation from  a  three-phase  system,  has  found  application  in 
synchronous  converters,  as  offering  a  higher  output  from  these 
machines,  and  a  symmetrical  eight-phase  system  proposed  for 
the  same  purpose. 

271.  A  characteristic  feature  of  the  symmetrical  n-phase  sys- 
tem is  that  under  certain  conditions  it  can  produce  a  rotating 
m.m.f.  of  constant  intensity. 

If  n  equal  magnetizing  coils  act  upon  a  point  under  equal 
angular  .displacements  in  space,  and  are  excited  by  the  n  e.m.fs. 
of  a  symmetrical  n-phase  system,  a  m.m.f.  of  constant  intensity 
is  produced  at  this  point,  whose  direction  revolves  synchronously 
with  uniform  velocity. 

Let 

n'     =  number  of  turns  of  each  magnetizing  coil. 
E     —  effective  value  of  impressed  e.m.f . 
I      =  effective  value  of  current. 
Hence, 

F  =  n'l  =  effective  m.m.f.  of  one  of  the  magnetizing  coils. 

Then  the  instantaneous  value  of  the  m.m.f.  of  the  coil  acting 

in  the  direction, ,  is 

n  ' 


=  n'l  V2  sin  (ft  -  ~ 


26 


402         ALTERNATING-CURRENT  PHENOMENA 

The  two  rectangular  space  components  of  this  m.m.f.  are 

.,  2iri 

/«     =  /<  cos  — 

f-         2iri    .       I          2iri\ 
=    n'l  V 2  cos  —  sin  \B I • 

and 

//'-/,  sin 


,  T    f-    ,     2  in    .      /         2-7r?A 
=  n  /v 2  sin sin   1/3 I- 

n  \  n   / 

Hence  the  m.m.f.  of  this  coil  can  be  expressed  by  the  symbolic 
formula 

/T    /-         (  n       2iri\  I        2iri   t    .   .    2iri\ 

/»  =  n  7\/2  sin  1/3 )  I  cos h  J  sin )  • 

Y          n  I  \          n  n  I 

Thus  the  total  or  resultant  m.m.f.  of  the  n  coils  displaced  under 
the  n  equal  angles  is 


v-r  /7     /S^-      •     /*        27rA/          27rt.  .2irA 

/  =  S*  /»  =  n7  V  2  2*  sin  (  0  —  -  )  I  cos  -  +  j  sin  - 

i  i  \  f*7\          n  n    I 

or,  expanded, 

rr    /sf    •      ov-/         »2iri    .     .    .     27ri          2irA 
/  =  n  /v  2     sin  |8  Zl  I   cos2  --  h  J  sin  --  cos  — 

I  i     \  n  n  n  I 

Q  "  .  /    .    2«        2Tt,  22H\\ 

—  cos  |8  2/»     sin  -  cos  --  h  J  sin2  - 

i     \          n  n  n  /j 

It  is,  however, 

'      92iri    ,  2iri          2irz         ,    /.  4^  47rA 

cos2  —  +  j  sin  —  cos  —    =  ±(1  +  cos  —  +  j  sin  —  j 


sm 


2irc         2irt   ,  ,2iri       j  A  4irl'  4ir*\ 

—  cos—  +jsm  2—  =  ^  (l  -  cos  —  -  3  sin  —  ) 


and,  since 

2<e2i  =  0,       S«e-  2i  =  0, 
i  i 

it  is, 


,  . 
/=  -—  —  -  (sm/3  -jcos/8); 


SYMMETRICAL  POLYPHASE  SYSTEMS          403 

or, 

,       nn'7  ,  .  •  N 

/  =  —^  (sin  0  -  j  cos  0) 

(sin0  —  j  cos0); 

the  symbolic  expression  of  the  m.m.f.  produced  by  the  n  circuits 
of  the  symmetrical  n-phase  system,  when  exciting  n  equal  mag- 
netizing coils  displaced  in  space  under  equal  angles. 
The  absolute  value  of  this  m.m.f.  is 

=  nn'I        nF     =  nFmax 
''  V2  ==  A/2  =        2 

Hence  constant  and  equal  — - ^=  times  the  effective  m.m.f.  of 

each  coil  or  ^  times  the  maximum  m.m.f.  of  each  coil. 

The  phase  of  the  resultant  m.m.f.  at  the  time  represented  by 
the  angle  0  is 

tan  6  =  —  cot  0;  hence  6  =  —  0  ~' 

That  is,  the  m.m.f.  produced  by  a  symmetrical  n-phase  system 
revolves  with  constant  intensity, 


V2 

and  constant  speed,  in  synchronism  with  the  frequency  of  the 
system;  and,  if  the  reluctance  of  the  magnetic  circuit  is  constant, 
the  magnetism  revolves  with  constant  intensity  and  constant 
speed  also,  at  the  point  acted  upon  symmetrically  by  the  n 
m.m.fs.  of  the  n-phase  system. 

This  is  a  characteristic  feature  of  the  symmetrical  polyphase 
system. 

272.  In  the  three-phase  system,  n  =  3,  FQ  =  1.5  Fmax,  where 
Fmax  is  the  maximum  m.m.f.  of  each  of  the  magnetizing  coils. 

In  a  symmetrical  quarter-phase  system,  n  =  4,  FQ  =  2  Fmax, 
where  Fmax  is  the  maximum  m.m.f.  of  each  of  the  four  magnet- 
izing coils,  or,  if  only  two  coils  are  used,  since  the  four-phase 
m.m.fs.  are  opposite  in  phase  by  two,  FQ  =  Fmax,  where  Fmax  is 
the  maximum  m.m.f.  of  each  of  the  two  magnetizing  coils  of  the 
quarter-phase  system. 


404         ALTERNATING-CURRENT  PHENOMENA 

While  the  quarter-phase  system,  consisting  of  two  e.m.fs.  dis- 
placed by  one-quarter  of  a  period,  is  by  its  nature  an  unsym- 
metrical  system,  it  shares  a  number  of  features — as,  for  instance, 
the  ability  of  producing  a  constant-resultant  m.m.f. — with  the 
symmetrical  system,  and  may  be  considered  as  one-half  of  a 
symmetrical  four-phase  system. 

Such  systems,  consisting  of  one-half  of  a  symmetrical  system, 
are  called  hemisymmetrical  systems. 


CHAPTER  XXX 

BALANCED  AND    UNBALANCED,  POLYPHASE  SYSTEMS 
273.  If  an  alternating  e.m.f., 

e  =  E\/2  sin  0, 

produces  a  current, 

i  =  7\/2  sin  (0  -  0), 

where  0  is  the  angle  of  lag,  the  power  is 

p  =  ei  =  2  EI  sin  0  sin  (0  -  0) 

=  EI  (cos  0  -  cos  (2  0  -  0)), 

and  the  average  value  of  power, 

P  =  EI  cos  0. 
Substituting  this,  the  instantaneous  value  of  power  is  found  as 

cos  (2  0  - 


Hence  the  power,  or  the  flow  of  energy,  in  an  ordinary  single- 
phase,  alternating-current  circuit  is  fluctuating,  and  varies  with 
twice  the  frequency  of  e.m.f.  and  current,  unlike  the  power  of  a 
continuous-current  circuit,  which  is  constant, 

p  =  ei. 
If  the  angle  of  lag,  0  =  0,  it  is, 

p  =  P(l  -  cos  20); 

hence  the  flow  of  energy  varies  between  zero  and  2  P,  where  P  is 
the  average  flow  of  energy  or  the  effective  power  of  the  circuit. 
If  the  current  lags  or  leads  the  e.m.f.  by  angle  0,  the  power 
varies  between 


cos    /  \         cos 

that  is,  becomes  negative  for  a  certain  part  of  each  half-wave. 
That  is,  for  a  time  during  each  half-wave,  energy  flows  back  into 

405 


406         ALTERNATING-CURRENT  PHENOMENA 

the  generator,  while  during  the  other  part  of  the  half-wave  the 
generator  sends  out  energy,  and  the  difference  Between  both  is 
the  effective  power  of  the  circuit. 
If  6  =  90°,  it  is 

p  =  -  El  sin  2  /3; 

that  is,  the  effective  power  P  =  0,  and  the  energy  flows  to  and 
fro  between  generator  and  receiving  circuit. 

Under  any  circumstances,  however,  the  flow  of  energy  in  the 
single-phase  system  is  fluctuating,  at  least  between  zero  and  a 
maximum  value,  frequently  even  reversing. 

274.  If  in  a  polyphase  system 

e\,  £2,  e3,   .    .    .    .    =  instantaneous  values  of  e.m.f.; 
iit  iz,  2*3,   .    .    .    .    =  instantaneous  values  of  current  pro- 
duced thereby, 

the  total  power  in  the  system  is 

p  =  eiii  +  eziz  +  03*3  +  •    •    •    • 
The  average  power  is 

P  =  EJi  cos  0i  +  EJz  cos  02  +  .    .    .    . 

The  polyphase  system  is  called  a  balanced  system,  if  the  flow 
of  energy 

p  =  eiii  +  e2iz 


is  constant,  and  it  is  called  an  unbalanced  system  if  the  flow  of 
energy  varies  periodically,  as  in  the  single-phase  system  ;  and  the 
ratio  of  the  minimum  value  to  the  maximum  value  of  power  is 
called  the  balance-factor  of  the  system. 

Hence  in  a  single-phase  system  on  non-inductive  circuit, 
that  is,  at  no-phase  displacement,  the  balance-factor  is  zero; 
and  it  is  negative  in  a  single-phase  system  with  lagging  or 
leading  current,  and  becomes  equal  to  —  1  if  the  phase  displace- 
ment is  90°  —  that  is,  the  circuit  is  wattless. 

275.  Obviously,  in  a  polyphase  system  the  balance  of  the 
system  is  a  function  of  the  distribution  of  load  between  the 
different  branch  circuits. 

A  balanced  system  in  particular  is  called  a  polyphase  system, 
whose  flow  of  energy  is  constant,  if  all  the  circuits  are  loaded 
equally  with  a  load  of  the  same  character,  that  is,  the  same  phase 
displacement. 


POLYPHASE  SYSTEMS  407 

276.  All  the  symmetrical  systems  from  the  three-phase  system 
upward  are  balanced  systems.  Many  unsymmetrical  systems 
are  balanced  systems  also. 

1.  Three-phase  system: 
Let 

€i  =  E  \/2  sin  0,  and  ii  =  I  \/2  sin  (0  —  0), 

€2  =  E  \/2  sin  (0  -  120),         i2  =  /  \/2  sin  (ft  -  6  -  120), 

e3  =  E  <\/2  sin  (0  -  240),         *,  =  I  \/2  sin  (]8  -  0  -  240), 

be  the  e.m.fs.  of  a  three-phase  system  and  the  currents  produced 
thereby. 

Then  the  total  power  is 

p  =  2  El  {sin  ft  sin  (0  -  0)  +  sin  (0  -  120)  sin  (0  -  0  -  120) 
+  sin  (0  -  240)  sin  (ft  -  6  -  240)  J 
=  3  El  cos  0  =  P,  or  constant. 

Hence  the  symmetrical  three-phase  system  is  a  balanced 
system. 

2.  Quarter-phase  system: 

Let     ei  =  E  \/2  sin  ft,       ii  =  I  \/2  sin  (ft  —  0), 
ez  =  E  \/2  cos  0,       i2  =  7  \/2  cos  (0  -  0) 

be  the  e.m.fs.  of  the  quarter-phase  system,   and   the   currents 
produced  thereby. 

This  is  an  unsymmetrical  system,  but  the  instantaneous  value 
of  power  is 

p  =  2  EI  {sin  0  sin  (ft  -  0)  +    cos  ft  cos  (ft  -  0)  j 
=  2  EI  cos  0  =  P,  or  constant. 

Hence  the  quarter-phase  system  is  an  unsymmetrical  balanced 
system. 

3.  The  symmetrical  n-phase  system,  with  equal  load  and  equal 
phase-displacement  in  all  n  branches,   is  a  balanced  system. 
For,  let 

ei  =  E \/2  sin  (ft — j   =  e.m.f . ; 

it  =  7\/2  sin  (0  —  0 )   =  current; 

\  71    / 


408         ALTERNATING-CURRENT  PHENOMENA 
the  instantaneous  value  of  power  is 


i 

=  2  El  S*  sin(0  -  ?^)  sin  (0  -  0  -  ^) 

f     »  "  /  47TA     1 

=  #/      S*    cos  0  -  S»'  cos  (2/3  -  0 I      ; 

{   i  i  \  »*•-.] 


or  p  =  n£7  cos  0  =  P,  or  constant. 

277.  An  unbalanced  polyphase  system  is  the  so-called  inverted 
three-phase  system,  derived  from  two  branches  of  a  three-phase 
system  by  transformation  by  means  of  two  transformers,  whose 
secondaries  are  connected  in  opposite  direction  with  respect  to 
their  primaries.     Such  a  system  takes  an  intermediate  position 
between   the   Edison   three-wire   system   and   the   three-phase 
system.     It  shares  with  the  latter  the  polyphase  feature,  and  with 
the  Edison  three- wire  system  the  feature  that  the  potential  differ- 
ence between  the  outside  wires  is  higher  than  between  middle  wire 
and  outside  wire. 

By  such  a  pair  of  transformers  the  two  primary  e.m.fs.  of  120° 
displacement  of  phase  are  transformed  into  two  secondary  e.m.fs., 
differing  from  each  other  by  60°.  Thus  in  the  secondary  circuit 
the  difference  of  potential  between  the  outside  wires  is  \/3  times 
the  difference  of  potential  between  middle  wire  and  outside  wire. 
At  equal  load  on  the  two  branches,  the  three  currents  are  equal, 
and  differ  from  each  other  by  120°,  that  is,  have  the  same  relative 
proportion  as  in  a  three-phase  system.  If  the  load  on  one 
branch  is  maintained  constant,  while  the  load  of  the  other  branch 
is  reduced  from  equality  with  that  in  the  first  branch  down  to 
zero,  the  current  in  the  middle  wire  first  decreases,  reaches  a 

minimum  value  of  -^-  =  0.866  of  its  original  value,  and  then 

increases  again,  reaching  at  no-load  the  same  value  as  at  full-load. 
The  balance  factor  of  the  inverted  three-phase  system  on  non- 
inductive  load  is  0.333. 

278.  In  Figs.  196  to  203  are  shown  the  e.m.fs.,  as  e  and  currents 
as  i  in  full  lines,  and  the  power  as  p  in  dotted  lines,  for  balance- 
factor,  0;  balance-factor,—  0.333;  balance-factor,  +  1;  balance- 
factor,  +  1;  balance-factor,  +  1;  balance-factor,  +  1;  balance- 
factor,  +  0.333,  and  balance-factor,  0. 


POLYPHASE  SYSTEMS  409 

279.  The  flow  of  energy  in  an  alternating-current  system  is  a 
most  important  and  characteristic  feature  of  the  system,  and  by 
its  nature  the  systems  may  be  classified  into: 

Monocyclic  systems,  or  systems  with  a  balance-factor  zero  or 
negative. 

Polycyclic  systems,  with  a  positive  balance-factor. 

Balance-factor  —1  corresponds  to  a  wattless  single-phase 
circuit,  balance-factor  zero  to  a  non-inductive  single-phase 
circuit,  balance-factor  + 1  to  a  balanced  polyphase  system. 

280.  In  polar  coordinates  the  flow  of  energy  of  an  alternating 
current  system  is  represented  by  using  the  instantaneous  value 
of  power  as  radius  vector,  with  the  angle,  j3,  corresponding  to 
the  time  as  amplitude,  one  complete  period  being  represented  by 
one  revolution. 

In  this  way  the  power  of  an  alternating-current  system  is 
represented  by  a  closed  symmetrical  curve,  having  the  zero  point 
as  quadruple  point.  In  the  monocyclic  systems  the  zero  point  is 
quadruple  nodal  point;  in  the  poly  cyclic  systems  quadruple 
isolated  point. 

Thus  these  curves  are  sextics. 

Since  the  flow  of  energy  in  any  single-phase  branch  of  the 
alternating-current  system  can  be  represented  by  a  sine  wave  of 
double  frequency, 

,   sin  (2  0  —  0)> 


P  = 


cos  0 


the  total  flow  of  energy  of  the  system  as  derived  by  the  addition 
of  the  powers  of  the  branch  circuits  can  be  represented  in  the 
form 

p  =P(1  +6  sin  (2/3  -  00)). 

This  is  a  wave  of  double  frequency  also,  with  e  as  amplitude  of 
fluctuation  of  power. 

This  is  the  equation  of  the  power  characteristics  of  the  system 
in  polar  coordinates. 

287.  To  derive  the  equation  in  rectangular  coordinates  we 

introduce  a  substitution  which  revolves  the  system  of  coordinates 
t\ 

by  an  angle,  -^,  so  as  to  make  the  symmetry  axes  of  the  power 
characteristic  the  coordinate  axes. 


P  = 


410         ALTERNATING-CURRENT  PHENOMENA 


hence, 

sin  (2  0-  90)  =  2  sin  (0  -  |°)  cos  (0  -  |°)  = 
substituted, 


or,  expanded, 

(a.2    +   02)8    _   P2(a.2    +   2/2    +   2  6^)2    =    0, 

the  sextic  equation  of  the  power  characteristic. 
Introducing 

a  =  (l-fe)P  =  maximum  value  of  power, 
6  =  (1  —  e)  P  =  minimum  value  of  power; 
we  have 

P       <*  +  *> 

2     ' 

a-b 
=  a  +  b' 

hence,  substituted,  and  expanded, 

(z2  +  y2)3  -  \  [a(x  +  yY  +  b(x  -  ^)2}2  =  0, 
the  equation  of  the  power  characteristic,  with  the  main  power 
axes,  a  and  b,  and  the  balance-factor,  — 

It  is  thus: 

Single-phase,  non-inductive  circuit,  p  =  P  (1  +  sin  2  6),  b  =  0, 
a  =  2P, 

(z2  +  t/2)3  -  P2(z  +  2/)4  =  0,  \  =  0. 

Single-phase  circuit,  60°  lag:  p  =  P  (1  +  2  sin  2  0),  6  -=  -  P, 
a  =  +  3  P, 

(X2   +  ^2)3    _   P2  (3.2   +3,2+4  Z2/)2    =    0,  b-     =    ~   g" 

Single-phase  circuit,  90°  lag:  p  =  El  sin  2  0, 

b  =  -  El,     a=  +EI, 
(a.2  +  ^2)3  _  4  (EI)*x*y*,  b-  =  -  1. 


POLYPHASE  SYSTEMS 


411 


Three-phase  non-inductive  circuit,  p  =  P,  6  =  1,  a  =  l, 


\ 


x2  +  y2  -  P2  =  0,  circle.    -  =  +  1. 


/v\ 


Fia.  196.— Single-phase,  non-inductive  circuit. 


FIG.  197.— Single-phase,  60°  lag. 


FIG.  19$.— Quarter-phase,  non-inductive  circuit. 
Three-phase  circuit,  60°  lag,  p=P,  6  =  l,a  =  l, 


+  y2  -  P2  =  0,  circle.    -  =  -f  1. 


412         ALTERNATING-CURRENT  PHENOMENA 
Quarter-phase  non-inductive  circuit,  p  =  P,  6  =  1,  a  =  1, 
x2  +  y2  -  P2  =  0,  circle.     -  =  +  1. 


FIG.  199.— Quarter-phase,  60°  lag. 


e  S  X    6 

•«* 

FIG.  200. — Three-phase,  non-inductive  circuit, 
e  v— ^  e. 


FIG.  201.— Three-phase,  60°  lag. 

Quarter-phase  circuit,  60°  lag,  p  =  P,  6  =  1,  a  =  l, 
x2  +  y2  -  P2  =  0,  circle.     -  =  +  1. 


POLYPHASE  SYSTEMS 


413 


FIG.  202. — Inverted  three-phase,  non-inductive  circuit. 


FIG.  203. — Inverted  three-phase,  60°  lag. 


414         ALTERNATING-CURRENT  PHENOMENA 
Inverted  three-phase  non-inductive  circuit, 


(x2  +  yY  -  P2  (x*  +  y*  +  xyY  =  0.     ~-  =  +    - 

Inverted  three-phase  circuit  60°  lag,  p  =  P(l  +  sin  2  0),  6  =  0, 
a  =  2P, 

z2         23-P2*  +  i4  =  0.         =  0. 


FIGS.  204  AND  205.—  Power 
characteristic  of  single-phase 
system,  at  0°  and  60°  lag. 


FIGS.  206  AND  207.—  Power 
characteristic  of  inverted  three- 
phase  system,  at  0°and  60°  lag. 


a  and  6  are  called  the  main  power  axes  of  the  alternating-cur- 
rent system,  and  the  ratio,  —  ,  is  the  balance-factor  of  the  system. 

282.  As  seen,  the  flow  of  energy  of  an  alternating-current  sys- 
tem is  completely  characterized  by  its  two  main  power  axes, 
a  and  6. 

The  power  characteristics  in  polar  coordinates,  corresponding 
to  the  Figs.  196,  197,  202  and  203  are  shown  in  Figs.  204,  205, 
206  and  207. 

The  balanced  quarter-phase  and  three-phase  systems  give  as 
polar  characteristics  concentric  circles. 


CHAPTER  XXXI 
INTERLINKED  POLYPHASE  SYSTEMS 

283.  In  a  polyphase  system  the  different  circuits  of  displaced 
phases,   which   constitute  the  system,   may  either  be  entirely 
separate  and  without  electrical  connection  with  each  other,  or 
they  may  be  connected  with  each  other  electrically,  so  that  a 
part  of  the  electrical  conductors  are  in  common  to  the  different 
phases,  and  in  this  case  the  system  is  called  an  interlinked  poly- 
phase system. 

Thus,  for  instance,  the  quarter-phase  system  will  be  called  an 
independent  system  if  the  two  e.m.fs.  in  quadrature  with  each 
other  are  produced  by  two  entirely  separate  coils  of  the  same, 
or  different,  but  rigidly  connected,  armatures,  and  are  connected 
to  four  wires  which  energize  independent  circuits  in  motors  or 
other  receiving  devices.  If  the  quarter-phase  system  is  derived 
by  connecting  four  equidistant  points  of  a  closed-circuit  drum 
or  ring-wound  armature  to  the  four  collector  rings,  the  system  is 
an  interlinked  quarter-phase  system. 

Similarly  in  a  three-phase  system.  Since  each  of  the  three 
currents  which  differ  from  each  other  by  one-third  of  a  period 
is  equal  to  the  resultant  of  the  other  two  currents,  it  can  be  con- 
sidered as  the  return  circuit  of  the  other  two  currents,  and  an 
interlinked  three-phase  system  thus  consists  of  three  wires  con- 
veying currents  differing  by  one-third  of  a  period  from  each 
other,  so  that  each  of  the  three  currents  is  a  common  return  of 
the  other  two,  and  inversely. 

284.  In  an  interlinked  polyphase  system  two  ways  exist  of 
connecting  apparatus  into  the  system. 

1.  The  star  connection,  represented  diagrammatically  in  Fig. 
208.  In  this  connection  the  n  circuits,  excited  by  currents  differ- 
ing from  each  other  by  -  of  a  period,  are  connected  with  their 

one  end  together  into  a  neutral  point  or  common  connection, 
which  may  either  be  grounded,  or  connected  with  other  corre- 
sponding neutral  points,  or  insulated. 

415 


416 


ALTERNATING-CURRENT  PHENOMENA 


In  a  three-phase  system  this  connection  is  usually  called  a  Y 
connection,  from  a  similarity  of  its  diagrammatical  representa- 
tion with  the  letter  F,  as  shown  in  Fig.  197. 


2.  The  ring  connection,  represented  diagrammatically  in  Fig. 
209,  where  the  n  circuits  of  the  apparatus  are  connected  with 
each  other  in  closed  circuit,  and  the  corners  or  points  of  connec- 
tion of  adjacent  circuits  connected  to  the  n  lines  of  the  polyphase 


FIG.  209. 

system.  In  a  three-phase  system  this  connection  is  called  the 
delta  (A)  connection,  from  the  similarity  of  its  diagrammatic 
representation  with  the  Greek  letter  delta,  as  shown  in  Fig.  193. 


INTERLINKED  POLYPHASE  SYSTEMS          417 

In  consequence  hereof  we  distinguish  between  star-connected 
and  ring-connected  generators,  motors,  etc.,  or  in  three-phase 
systems  Y-connected  and  A-connected  apparatus. 

285.  Obviously,  the  polyphase  system  as  a  whole  does  not 
differ,  whether  star  connection  or  ring  connection  is  used  in  the 
generators  or  other  apparatus;  and  the  transmission  line  of  a 
symmetrical  n-phase  system  always  consists  of  n  wires  carrying 
currents  of  equal  strength,  when  balanced,  differing  from  each 

other  in  phase  by  —  of  a  period.     Since  the  line  wires  radiate 

from  the  n  terminals  of  the  generator,  the  lines  can  be  considered 
as  being  in  star  connection. 

The  circuits  of  all  the  apparatus,  generators,  motors,  etc.,  can 
either  be  connected  in  star  connection,  that  is,  between  one  line 
and  a  neutral  point,  or  in  ring  connection,  that  is,  between  two 
adjacent  lines. 

In  general  some  of  the  apparatus  will  be  arranged  in  star  con- 
nection, some  in  ring  connection,  as  the  occasion  may  require. 
.  286.  In  the  same  way  as  we  speak  of  star  connection  and  ring 
connection  of  the  circuits  of  the  apparatus,  the  terms  star  voltage 
and  ring  voltage,  star  current  and  ring  current,  etc.,  are  used, 
whereby  as  star  voltage  or  in  a  three-phase  circuit  Y  voltage,  the 
potential  difference  between  one  of  the  lines  and  the  neutral 
point,  that  is,  a  point  having  the  same  difference  of  potential 
against  all  the  lines,  is  understood;  that  is,  the  voltage  as  meas- 
ured by  a  voltmeter  connected  into  star  or  Y  connection.  By 
ring  or  delta  voltage  is  understood  the  difference  of  potential 
between  adjacent  lines,  as  measured  by  a  voltmeter  connected 
between  adjacent  lines,  in  ring  or  delta  connection. 

In  the  same  way  the  star  or  Y  current  is  the  current  in  a  cir- 
cuit from  one  line  to  a  neutral  point;  the  ring  or  delta  current, 
the  current  in  a  circuit  from  one  line  to  the  next  line. 

The  current  in  the  transmission  line  is  always  the  star  or  Y 
current,  and  the  potential  difference  between  the  line  wires,  the 
ring  or  delta  voltage. 

Since  the  star  voltage  and  the  ring  voltage  differ  from  each 
other,  apparatus  requiring  different  voltages  can  be  connected 
into  the  same  polyphase  mains,  by  using  either  star  or  ring 
connection. 

287.  If  in  a  generator  with  star-connected  circuits,  the  e.m.f. 
per  circuit  =  Et  and  the  common  connection  or  neutral  point 

27 


418         ALTERNATING-CURRENT  PHENOMENA 

is  denoted  by  zero,  the  voltages  of  the  n  terminals  are 

E,  eE,  JE  .    .    ,    .   en~lE', 
or  in  general,  e\Z£, 

at  the  iih  terminal,  where, 

i  =  0,  1,  2   .    .    .    .   n  —  1,    e  =  cos  ---  h  j  sin  —     =  \/l. 

Tb  T\J 

Hence  the  e.m.f  .  in  the  circuit  from  the  iih  to  the  kih  terminal  is 

Eki  =  ek  E  -  CE  =  (ek  -  ^E. 
The  e.m.f.  between  adjacent  terminals  i  and  i  +  1  is 


In  a  generator  with  ring-connected  circuits,  the  e.m.f.  per 
circuit 

JE, 

is  the  ring  e.m.f.,  and  takes  the  place  of 


while  the  e.m.f.  between  terminal  and  neutral  point,  or  the  star 
e.m.f.,  is 


Hence  in  a  star-connected  generator  with  the  e.m.f.  E  per 
circuit,  it  is: 

star  e.m.f.,  c*  E, 

ring  e.m.f  .,  ei  (e  —  1)E, 

e.m.f.  between  terminal  i  and  terminal  k,  (ek  —  e^E. 

In  a  ring-connected  generator  with  the  e.m.f.,  E,  per  circuit, 
it  is 

star  e.m.f., 


,, 

€  —   1  • 

ring  e.m.f.,  t{E, 

e.m.f.  between  terminals  i  and  k, 


In  a  star-connected  apparatus,  the  e.m.f.  and  the  current  per 


INTERLINKED  POLYPHASE  SYSTEMS          419 

circuit  have  to  be  the  star  e.m.f.  and  the  star  current.  In  a 
ring-connected  apparatus  the  e.m.f.  and  current  per  circuit  have 
to  be  the  ring  e.m.f.  and  ring  current. 

In  the  generator  of  a  symmetrical  polyphase  system,  if 

e{E  are  the  e.m.fs.  between  the  n  terminals  and  the  neutral 
point,  or  star  e.m.fs. 

Ii  =  the  currents  issuing  from  terminals  i  over  a  line  of  the 
impedance,  Zi  (including  generator  impedance  in  star  connec- 
tion), we  have 

voltage  at  end  of  line  i, 

e*E  -  ZJi, 

and  difference  of  potential  between  terminals  k  and  i 

(ek  -  e^E  -  (Zklk  -  Zili), 

where  /<  is  the  star  current  of  the  system,  Zi  the  star  impedance. 
The  ring  voltage  at  the  end  of  the  line  between  terminals  i 

and  k  is  EM,  and 

Eik  =  —  Eki> 

If  now  lik  denotes  the  current  from  terminal  i  to  terminal  k, 
and  Zik  impedance  of  the  circuit  between  terminal  i  and  ter- 
minal k,  where 

lik     =    —   Iki, 

Zik    =    Zki, 

we  have  Eik  =  Ziklik. 

If  lio  denotes  the  current  in  the  circuit  from  terminal  i  to  a 
ground  or  neutral  point,  and  Zt-o  is  the  impedance  of  this  circuit 
between  terminal  i  and  neutral  point,  it  is 

E io    ==   f^E   —    Z  il  i    =    Z  iol  io- 

288.  We  have  thus,  by  Ohm's  law  and  Kirchoff's  law: 

If  eiE  is  the  e.m.f.  per  circuit  of  the  generator,  between  the 
terminal,  i,  and  the  neutral  point  of  the  generator,  or  the  star 
e.m.f. 

Ii  =  the  current  at  the  terminal,  i,  of  the  generator,  or  the 
star  current. 

Zi  =  the  impedance  of  the  line  connected  to  a  terminal,  ?',  of 
the  generator,  including  generator  impedance. 


420         ALTERNATING-CURRENT  PHENOMENA 

Ei  =  the  e.m.f.  at  the  end  of  line  connected  to  a  terminal,  i, 
of  the  generator. 

Eik  =  the  difference  of  potential  between  the  ends  of  the  lines, 
i  and  fe. 

Iik   =  the  current  from  line  i  to  line  k. 

Zik  =  the  impedance  of  the  circuit  between  lines  i  and  k. 

/to,  /too  .  .  .  .  =  the  current  from  line  i  to  neutral  points 
0,  00,  '..'.. 

Zio,  Zioo  .  .  .  .  =  the  impedance  of  the  circuits  between 
line  i  and  neutral  points  0,  00,  .... 

Then: 

177T  T7T  T  T  77  __      f7  T  ~f 

..    Hi  \k    —        ~"   J^kij    *•  ik    ~        "~   J-ki    ^  ik    —    **£*)    -*  to    —        ~   I0iy 

A/  {o    —    fjoi*   GT/C/» 

2.  /?»    = 

3.  E»    = 

4.  Eifc  =  Efc-  Et  =  (e*  -  eOE  -  (ZJfc  -  Zi/t-). 

n 

6.  /,    =  2*/tfc. 

0    * 

7.  If  the  neutral  point  of  the  generator  does  not  exist,  as  in 
ring  connection,  or  is  insulated  from  the  other  neutral  points : 

n 

2*1  i      =  0 

1    • 

S'/i.   =0; 

1 

n 

S</,-oo  =  0,  etc 

Where  0,  00,  etc.,  are  the  different  neutral  points  which  are 
insulated  from  each  other. 

If  the  neutral  point  of  the  generator  and  all  the  other  neutral 
points  are  grounded  or  connected  with  each  other,  we  have, 


1  1 


1    • 


INTERLINKED  POLYPHASE  SYSTEMS          421 

If  the  neutral  point  of  the  generator  or  other  neutral  points 
are  grounded,  the  system  is  called  a  grounded  system.  If  the 
neutral  points  are  not  grounded,  the  system  is  an  insulated  poly- 
phase system,  and  an  insulated  polyphase  system  with  equalizing 
return,  if  all  the  neutral  points  are  connected  with  each  other. 

8.  The  power  of  the  polyphase  system  is 


n 


S»'  €1EI  a  cos  0i  at  the  generator, 


P  —  2*2*  Eiklik  cos  6ik  in  the  receiving  circuits. 


CHAPTER  XXXII 
TRANSFORMATION  OF  POLYPHASE  SYSTEMS 

289.  In  transforming  one  polyphase  system  into  another  poly- 
phase system,  it  is  obvious  that  the  primary  system  must  have 
the  same  flow  of  energy  as  the  secondary  system,  neglecting 
losses  in  transformation,  and  that  consequently  a  balanced  sys- 
tem will  be  transformed  again  into  a  balanced  system,  and  an 
unbalanced  system  into  an  unbalanced  system  of  the  same  bal- 
ance-factor, since  the  transformer  is  not  able  to  store  energy, 
and  thereby  to  change  the  nature  of  the  flow  of  energy.     The 
energy  stored  as  magnetism  amounts  in  a  well-designed  trans- 
former only  to  a  very  small  percentage  of  the  total  energy.     This 
shows  the  futility  of  producing  symmetrical  balanced  polyphase 
systems  by  transformation  from  the  unbalanced  single-phase 
system  without  additional  apparatus  able  to  store  energy  effi- 
ciently, as  revolving  machinery,  etc. 

Since  any  e.m.f.  can  be  resolved  into,  or  produced  by,  two 
components  of  given  directions,  the  e.m.f.  of  any  polyphase  sys- 
tem can  be  resolved  into  components  or  produced  from  compon- 
ents of  two  given  directions.  This  enables  the  transformation 
of  any  polyphase  system  into  any  other  polyphase  system  of  the 
same  balance-factor  by  two  transformers  only. 

290.  Let  Ei,  Ez,  Ez  .    .    .    .  be  the  e.m.f s.  of  the  primary  sys- 
tem which  shall  be  transformed  into 

E'iy  E'z,  E'z   ....   the  e.m.fs.  of  the  secondary  system. 

Choosing  two  magnetic  fluxes,  $>  and  3>,  of  different  phases, 
as  magnetic  circuits  of  the  two  transformers,  which  generate  the 
e  m.fs.,  e  and  e,  per  turn,  by  the  law  of  parallelogram  the  e.m.fs., 
El,  EZ,  ....  can  be  resolved  into  two  components,  ~E\  and 
Ei,  Ez  and  E?2,  ....  of  the  phases,  ~e  and  ~e. 
Then_ 

Ely  Ez,   ....  are  the  counter  e.m.fs.  which  have  to  be  gen- 
_  erated  in  the  primary  circuits  of  the  first  transformer; 
Ely  THz,   ....  the  counter  e.m.fs.  which  have  to  be  generated 

in  the  primary  circuits  of  the  second  transformer. 

422 


TRANSFORMATION  OF  POLYPHASE  SYSTEMS  423 

Hence 

7^     7^ 

— »  ---  .    .    .    .  are  the  numbers  of  turns  of  the  primary  coils  of 
e     e 

the  first  transformer. 
Analogously 

7T      ~W  ' 

-=-»  —    .    .    .  .  are  the  numbers  of  turns  of  the  primary  coils  in 

e      e 

the  second  transformer. 

In  the  same  manner  as  the  e.m.fs.  of  the  primary  system  have 
been  resolved  into  components  in  phase  with  e  and  e,  the  e.m.fs.  of 
the  secondary  system,  E'i,  E'%,  ..:'...  are  produced  from  com- 
ponents, E'i,  and  E'i,  E'*,  and  E'z  .  .  .  .  in  phase  with  e  and 
e,  and  give  as  numbers  of  secondary  turns — 

-=-»  -=-   ...    .in  the  first  transformer: 

e       e 

Tjlf  EV 

-*->  -s—   ...     .in  the  second  transformer. 

e      e 

That  means  each  of  the  two  transformers,  m  and  ra,  contains  in 
general  primary  turns  of  each  of  the  primary  phases,  and  second- 
ary turns  of  each  of  the  secondary  phases.  Loading  now  the 
secondary  polyphase  system  in  any  desired  manner,  correspond- 
ing to  the  secondary  currents,  primary  currents  will  exist  in  such 
a  manner  that  the  total  flow  of  energy  in  the  primary  polyphase 
system  is  the  same  as  the  total  flow  of  energy  in  the  secondary 
system,  plus  the  loss  of  power  in  the  transformers. 

291.  As  an  instance  may  be  considered  the  transformation  of 
the  symmetrical  balanced  three-phase  system, 

E  sin  j8,     E  sin  (0  -  120),     E  sin  (0  -  240), 
into  an  unsymmetrical  balanced  quarter-phase  system, 
E'  sin  ft     E'  sin  (0  -  90). 

Let  the  magnetic  flux  of  the  two  transformers  be  chosen  in  quad- 
rature 

$  cos  j8  and  &  cos  (0  —  90). 

Then  the  e.m.fs.  generated  per  turn  in  the  transformers  are 
e  sin  0  and  e  sin  (ft  —  90) ; 


424         ALTERNATING-CURRENT  PHENOMENA 

hence,  in  the  primary  circuit  the  first  phase,  E  sin  0,  will  give,  in 

E 

the  first  transformer,  —  primary  turns  ;  in  the  second  transformer, 

6 

0  primary  turns. 

The  second  phase,  E  sin  (/3  —  120),  will  give,  in  the  first  trans- 

-  E 
former,  -~  —  primary  turns  ;  in  the  second  transformer,  —  ~ 

primary  turns. 

The  third  phase,  E  sin  (j3  —  240),  will  give,  in  the  first  trans- 

J7I  _     Tjl    \.s 

former,  -=  —  -  primary  turns  ;  in  the  second  transformer, 


, 

Z  6  Z  e 

primary  turns. 

In  the  secondary  circuit  the  first  phase,  E'  sin  /3,  will  give  in 

T?r 

the  first  transformer:  --  secondary  turns;  in  the  second  trans- 

e 

former:  0  secondary  turns. 

The  second  phase:  E'  sin  (/3  —  90)  will  give  in  the  first  trans- 

T?' 

former  :  0  secondary  turns  ;  in  the  second  transformer,  —  second- 

ary turns. 
Or,  if 

E  =  5000,  E'  =  100,  e  =  10. 

PRIMARY  SECONDARY 

1st.        2d.         3d.  1st.  2d. 

First  transformer              +500     -250     -250  10       0 

Second  transformer                 0     +433    -433  0     10  turns. 

Using  auto  transformer  connection  in  the  three-phase  primaries 
of  the  first  transformer,  that  is,  using  as  coils  of  the  second  and 
the  third  phase  the  two  halves  of  the  coil  of  the  first  phase,  this 
gives  the  well  known  T-connection  of  three-phase-quarter-phase 
transformation. 

That  means  : 

Any  balanced  polyphase  system  can  be  transformed  by  two 
transformers  only,  without  storage  of  energy,  into  any  other  balanced 
polyphase  system. 

Or  more  generally  stated: 

Any  polyphase  system  can  be  transformed  by  two  transformers 
only,  without  storage  of  energy,  into  any  other  polyphase  system 
of  the  same  balance  factor. 


TRANSFORMATION  OF  POLYPHASE  SYSTEMS  425 

292.  Some  of  the  more  common  methods  of  transformation 
between  polyphase  systems  are: 

1.  The  delta-Y  connection  of  transformers  between  three-phase 
systems,  shown  in  Fig.  210.  One  side  of  the  transformers  is 
connected  in  delta,  the  other  in  Y.  This  arrangement  becomes 
necessary  for  feeding  four-wire  three-phase  secondary  distribu- 
tions. The  Y  connection  of  the  secondary  allows  the  bringing 
out  of  a  neutral  wire,  while  the  delta  connection  of  the  primary 
maintains  the  balance,  in  regard  to  the  voltage  between  the 
phases  at  unequal  distribution  of  load. 

The  delta-Y  connection  of  step-up  transformers  is  frequently 
used  in  long-distance  transmissions,  to  allow  grounding  of  the 
high-potential  neutral.  Under  certain  conditions — which  there- 
fore have  to  be  guarded  against — it  is  liable  to  induce  excessive 
voltages  by  resonance  with  the  line  capacity. 


FIG.  210. 

The  reverse  thereof,  or  the  Y-delta  connection,  is  undesirable 
on  unbalanced  load,  since  it  gives  what  has  been  called  a  "  float- 
ing neutral;"  the  three  primary  Y  voltages  do  not  remain  even 
approximately  constant,  at  unequal  distribution  of  load  on  the 
secondary  delta,  but  the  primary  voltage  corresponding  to  the 
heavier  loaded  secondary,  and,  therefore,  also  the  corresponding- 
secondary  voltage,  collapses.  Thereby  the  common  connection 
of  the  primary  shifts  toward  one  corner  of  the  e.m.f.  triangle, 
away  from  the  center  of  the  triangle,  and  may  even  fall  outside 
of  the  triangle.  As  result  thereof  the  secondary  triangle  becomes 
very  greatly  distorted  even  at  moderate  inequality  of  load,  and 
the  system  thus  loses  all  ability  to  maintain  constant  voltage  at 
unequal  distribution  of  load,  that  is,  becomes  inoperative.  In 
high-potential  systems  in  this  case  excessive  voltages  may  be 
induced  by  resonance  with  the  line  capacity. 

For  instance,  if  only  one  phase  of  the  secondary  triangle  is 


426         ALTERNATING-CURRENT  PHENOMENA 

loaded,  the  other  two  unloaded,  the  primary  current  of  the 
loaded  phase  must  return  over  the  other  two  transformers,  which, 
at  open  secondaries,  act  as  very  high  reactances,  thus  limiting 
the  current  and  consuming  practically  all  the  voltage,  and  the 
loaded  primary,  and  thus  its  secondary,  receive  practically  no 
voltage. 

Y-delta  connection  is  satisfactory  if  the  secondary  load  is 
balanced,  as  induction — or  synchronous  motors,  or  if  the  primary 
neutral  is  connected  with  the  generator  neutral  or  the  secondary 
neutral  of  step-up  transformers  in  which  the  primaries  are  con- 
nected in  delta,  and  the  unbalanced  current  can  return  over  the 
neutral.  If  with  Y-delta  connection,  in  addition  to  an  un- 
balanced load,  the  secondary  carries  polyphase  motors,  the 
motors  take  different  currents  in  the  different  phases,  so  tha\t 
the  total  current  is  approximately  the  same  in  all  three  phases. 
That  is,  the  motors  act  as  phase  converters,  and  so  partially 
restore  the  balance  of  the  system. 

2.  The  delta-delta  connection  of  transformers  between  three- 
phase  systems,  in  which  primaries  as  well  as  secondaries  are  con- 
nected in  the  same  manner  as  the  primaries  in  Fig.  210. 

Since  in  this  system  each  phase  is  transformed  by  a  separate 
transformer,  the  voltages  of  the  system  remain  balanced  even  at 
unbalanced  load,  within  the  limits  of  voltage  variation  due  to 
the  internal  self-inductive  impedance  (or  short-circuit  impedance) 
of  the  transformers — which  is  small,  while  the  exciting  impedance 
(or  open-circuit  impedance)  of  the  transformers,  which  causes 
the  unbalancing  in  the  Y-delta  connection  above  discussed  is 
enormous. 

3.  Y-F  connection  of  transformers  between  three-phase  sys- 
tems.    Primaries  and  secondaries  connected  as  the  secondaries 
in  Fig.  210. 

In  this  case,  if  the  neutral  is  not  fixed  by  connection  with  a 
fixed  neutral,  either  directly  or  by  grounding  it,  the  neutral  also 
is  floating,  and  so  abnormal  voltages  may  be  produced  between 
the  lines  and  the  neutral,  without  appearing  in  the  voltages  be- 
tween the  lines,  and  may  lead  to  disruptive  effects,  or  to  over- 
heating of  the  transformers,  so  that  this  connection  is  not  an 
entirely  safe  one. 

Where  in  transformer  connections  in  polyphase  systems,  a 
neutral  or  common  connection  of  the  transformers  exists,  care 
must,  therefore,  be  taken  to  have  this  neutral  a  fixed  voltage 


TRANSFORMATION  OF  POLYPHASE  SYSTEMS  427 


point,  irrespective  of  the  variation  of  the  load  or  its  distribution, 
which  may  occur;  otherwise  harmful  phenomena  may  result  from 
a  " floating"  or  " unstable"  neutral. 

In  connections  (2)  and  (3),  the  secondary-e.m.f.  triangle  is  in 
phase  with  the  primary-e.m.f.  triangle,  while  in  (1)  it  is  displaced 
therefrom  by  30°.  Therefore,  even  if  the  voltages  are  equal,  con- 
nection (1)  cannot  be  operated  in  parallel  with  (2)  or  (3),  but  (2) 


FIG.  211. 

and  (3)  can  be  operated  in  parallel  with  each  other,  and  with  the 
connections  (4)  and  (5),  provided  that  the  voltages  are  correct. 

4.  The  V  connection  or  open  delta  connection  of  transformers 
between  three-phase  systems,  consists  in  using  two  sides  of  the 
triangle  only,  as  shown  in  Fig.  211.  This  arrangement  has  the 
disadvantage  of  transforming  one  phase  by  two  transformers  in 
series,  hence  is  less  efficient,  and  is  liable  to  unbalance  the  system 


\ 

/ 

\           5 

:           / 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 


FIG.  212. 

by  the  internal  impedance  of  the  transformers.  It  is  convenient 
for  small  powers  at  moderate  voltage,  since  it  requires  only  two 
transformers,  but  is  dangerous  in  high  potential  circuits,  being 
liable  to  produce  destructive  voltages  by  its  electrostatic  un- 
balancing. 

5.  The  main  and  teaser,  or  T  connection  of  transformers  be- 
tween three-phase  systems,  is  shown  in  Fig.  212.     One  of  the 


428         ALTERNATING-CURRENT  PHENOMENA 


two  transformers  is  wound  for  — x—  times  the  voltage  of  the  other 

Zi 

(the  altitude  of  the  equilateral  triangle),  and  connected  with  one 
of  its  ends  to  the  center  of  the  other  transformer.  From  the 
point  one-third  inside  of  the  teaser  transformer,  a  neutral  wire 
can  be  brought  out  in  this  connection. 

6.  The   monocyclic   connection,   transforming   between   three- 


FIG.  213. 

phase  and  inverted  three-phase  or  polyphase  monocyclic,  by  two 
transformers,  the  secondary  of  one  being  reversed  regarding  its 
primary,  as  shown  in  Fig.  213. 

7.  The  L  connection  for  transformation  between  quarter-phase 
and  three-phase  as  described  in  the  example,  §291. 

8.  The  T  connection  of  transformation  between  quarter-phase 
and  three-phase,  as  shown  in  Fig.  214.     The  quarter-phase  sides 


FIG.  214. 

of  the  transformers  contain  two  equal  and  independent  (or  inter- 
linked) coils,  the  three-phase  sides  two  coils  with  the  ratio  of 

turns,  1  -. — JT-,  connected  in  T. 

m 

9.  The  double  delta  connection  of  transformation  from  three- 
phase  to  six-phase,  shown  in  Fig.  215.  Three  transformers,  with 
two  secondary  coils  each,  are  used,  one  set  of  secondary  coils 
connected  in  delta,  the  other  set  in  delta  also,  but  with  reversed 


TRANSFORMATION  OF  POLYPHASE  SYSTEMS  429 


terminals,  so  as  to  give  a  reversed  e.m.f.  triangle.     These  e.m.fs. 
thus  give  topographically  a  six-cornered  star. 


A 


i'      UmlLMaAfli 
RP1 


Y 


TI       I 


n  m  I 


I        2 
FIG.  215. 


10.  The  double  Y  connection  or  diametrical  connection  of  trans- 
formation from  three-phase  to  six-phase,  shown  in  Fig.  216.     It 


FIG.  216. 

is  analogous  to  (7),  the  delta  connection  merely  being  replaced 
by  the  Y  connection.  The  neutrals  of  the  two  Y's  may  be  con- 
nected together  and  to  an  external  neutral  if  desired. 


/  A4A  m 


L 


I I 


100 


3' 


FIG.  217. 


The  primaries  in  9  and  10  may  be  connected  either  delta  or  Y, 
and  in  the  latter  case  a  floating  neutral  must  be  guarded  against. 


430         ALTERNATING-CURRENT  PHENOMENA 

11.  The  double  T  connection  of  transformation  from  three- 
phase  to  six-phase,  shown  in  Fig.  216.  Two  transformers  are 
used  with  two  secondary  coils  which  are  T-connected,  but  one 
with  reversed  terminals.  This  method  also  allows  a  secondary 
neutral  to  be  brought  out. 

293.  Transformation  with  a  change  of  the  balance-factor  of 
the  system  is  possible  only  by  means  of  apparatus  able  to  store 
energy,  since  the  difference  of  energy  between  primary  and 
secondary  circuit  has  to  be  stored  at  the  time  when  the  secondary 
power  is  below  the  primary,  and  returned  during  the  time  when 
the  primary  power  is  below  the  secondary.  The  most  efficient 
storing  device  of  electric  energy  is  mechanical  momentum  in  re- 
volving machinery.  It  has,  however,  the  disadvantage  of  re- 
quiring attendance;  fairly  efficient  also  are  condensive  and  in- 
ductive reactances,  but,  as  a  rule,  they  have  the  disadvantage  of 
not  giving  constant  potential. 


CHAPTER  XXXIII 
EFFICIENCY  OF  SYSTEMS 

294.  In  electric  power  transmission  and  distribution,  wherever 
the  place  of  consumption  of  the  electric  energy  is  distant  from 
the  place  of  production,  the  conductors  which  carry  the  current 
are  a  sufficiently  large  item  to  require  consideration,  when  decid- 
ing which  system  and  what  potential  is  to  be  used. 

In  general,  in  transmitting  a  given  amount  of  power  at  a  given 
loss  over  a  given  distance,  other  things  being  equal,  the  amount 
of  copper  required  in  the  conductors  is  inversely  proportional  to 
the  square  of  the  potential  used.  Since  the  total  power  trans- 
mitted is  proportional  to  the  product  of  current  and  e.m.f.,  at  a 
given  power,  the  current  will  vary  inversely  proportionally  to 
the  e.m.f.,  and  therefore,  since  the  loss  is  proportional  to  the 
product  of  current-square  and  resistance,  to  give  the  same  loss  the 
resistance  must  vary  inversely  proportional  to  the  square  of  the 
current,  that  is,  proportional  to  the  square  of  the  e.m.f.;  and 
since  the  amount  of  copper  is  inversely  proportional  to  the  resist- 
ance, other  things  being  equal,  the  amount  of  copper  varies  in- 
versely proportional  to  the  square  of  the  e.m.f.  used. 

This  holds  for  any  system. 

Therefore  to  compare  the  different  systems,  as  two-wire  single- 
phase,  single-phase  three-wire,  three-phase  and  quarter-phase, 
equality  of  the  potential  must  be  assumed. 

Some  systems,  however,  as,  for  instance,  the  Edison  three- 
wire  system,  or  the  inverted  three-phase  system,  have  different 
potentials  in  the  different  circuits  constituting  the  system,  and 
thus  the  comparison  can  be  made  either — 

1st.  On  the  basis  of  the  maximum  potential  difference  between 
any  two  conductors  of  the  system ;  or 

2nd.  On  the  basis  of  the  maximum  potential  difference  between 
any  conductor  of  the  system  and  the  ground ;  or 

3rd.  On  the  basis  of  the  minimum  potential  difference  in  the 
system,  or  the  potential  difference  per  circuit  or  phase  of  the 
system. 

•      431 


432         ALTERNATING-CURRENT  PHENOMENA 

In  low-potential  circuits,  as  secondary  networks,  where  the 
potential  is  not  limited  by  the  insulation  strain,  but  by  the 
potential  of  the  apparatus  connected  into  the  system,  as  incan- 
descent lamps,  the  proper  basis  of  comparison  is  equality  of  the 
potential  per  branch  of  the  system,  or  per  phase. 

On  the  other  hand,  in  long-distance  transmissions  where  the 
potential  is  not  restricted  by  any  consideration  of  apparatus 
suitable  for  a  certain  maximum  potential  only,  but  where  the 
limitation  of  potential  depends  upon  the  problem  of  insulating 
the  conductors  against  disruptive  discharge,  the  proper  com- 
parison is  on  the  basis  of  equality  of  the  maximum  difference 
of  potential;  that  is,  equal  maximum  dielectric  strain  on  the 
insulation. 

In  this  case,  the  comparison  voltage  may  be  either  the  poten- 
tial difference  between  any  two  conductors  of  the  system,  or 
it  may  be  the  potential  difference  between  any  conductor  of 
the  system  and  the  ground,  depending  on  the  character  of  the 
circuit. 

The  dielectric  stress  is  from  conductor  to  conductor,  or  be- 
tween any  two  conductors,  in  a  system  which  is  insulated  from 
the  ground,  as  is  mostly  the  case  in  medium  voltage  overhead 
transmissions,  and  frequently  in  underground  cables. 

In  an  ungrounded  cable  system,  in  which  all  the  conductors 
are  enclosed  in  the  same  cable,  the  insulation  stress  is  mainly 
from  conductor  to  conductor,  and  this  therefore  is  the  basis  of 
comparison.  But  even  in  an  underground  cable  system  with 
grounded  neutral,  as  very  commonly  used,  a  direct  path  exists 
from  conductor  to  conductor  inside  of  the  cables,  for  a  disrup- 
tive voltage,  and  the  comparison  of  systems,  therefore,  has  to  be 
made,  in  this  case,  on  the  basis  of  maximum  potential  difference 
between  conductors  as  well  as  between  conductor  and  ground. 

In  an  ungrounded  overhead  system,  the  disruptive  stress  is 
from  conductor  to  ground  and  back  from  ground  to  conductor. 
If  the  system  is  of  considerable  extent — as  is  the  case  where  high 
voltages  of  serious  disruptive  strength  have  to  be  considered — 
the  neutral  of  the  system  is  maintained  at  approximate  ground 
potential  by  the  capacity  of  the  system,  and  the  normal  voltage 
stress  from  conductor  to  ground  therefore  is  that  from  conductor 
to  neutral,  that  is,  the  same  as  in  a  system  with  grounded  neutral, 
and  the  basis  of  comparison  then  is  the  voltage  from  line  to 
ground,  and  not  between  lines.  Since,  however,  one  conductor 


EFFICIENCY  OF  SYSTEMS  433 

of  the  system  may  temporarily  ground,  if  it  is  required  to  main- 
tain operation  even  with  one  conductor  of  the  system  grounded, 
the  voltage  between  conductors  must  be  the  basis  of  comparison, 
since  with  one  conductor  grounded,  the  disruptive  stress  between 
the  other  conductors  and  ground  is  the  potential  difference  be- 
tween the  conductors  of  the  system. 

In  an  overhead  system  with  grounded  neutral,  frequently  used 
for  transmission  systems  of  very  high  voltage,  or  in  general  in  a 
grounded  system,  the  disruptive  stress  is  that  due  to  the  potential 
difference  between  conductor  and  ground  or  neutral,  and  this 
then  is  the  basis  of  comparison. 

In  moderate-potential  power  circuits,  in  considering  the  danger 
to  life  from  live  wires  entering  buildings  or  otherwise  accessible, 
the  comparison  on  the  basis  of  maximum  potential  also  appears 
appropriate. 

Thus  the  comparison  of  different  systems  of  long-distance 
transmission  at  high  potential  or  power  distribution  for  motors 
is  to  be  made  on  the  basis  of  equality  of  the  maximum  difference 
of  potential  existing  in  the  system ;  the  comparison  of  low-poten- 
tial distribution  circuits  for  lighting  on  the  basis  of  equality  of 
the  minimum  difference  of  potential  between  any  pair  of  wires 
connected  to  the  receiving  apparatus. 

295.  1st.  Comparison  on  the  basis  of  equality  of  the  minimum 
difference  of  potential,  in  low-potential  lighting  circuits: 

In  the  single-phase,  alternating-current  circuit,  if  e  =  e.m.f., 
i  =  current,  r  =  resistance  per  line,  the  total  power  is  =  ei}  the 
loss  of  power,  2  i*r. 

Using,  however,  a  three- wire  system:  the  potential  between 
outside  wires  and  neutral  being  given  equal  to  e,  the  potential 
between  the  outside  wires  is  equal  to  2  e}  that  is,  the  distribution 
takes  place  at  twice  the  potential,  or  only  one-fourth  the  copper 
is  needed  to  transmit  the  same  power  at  the  same  loss,  if,  as  it  is 
theoretically  possible,  the  neutral  wire  has  no  cross-section.  If, 
however,  the  neutral  wire  is  made  of  the  same  cross-section  as 
each  of  the  outside  wires,  three-eighths  as  much  copper  as  in  the 
two- wire  system  is  needed;  if  the  neutral  wire  is  one-half  the 
cross-section  of  each  of  the  outside  wires,  five-sixteenths  as  much 
copper  is  needed.  Obviously,  a  single-phase,  five-wire  system 
will  be  a  system  of  distribution  at  the  potential,  4  e,  and  there- 
fore require  only  one-sixteenth  of  the  copper  of  the  single-phase 
system  in  the  outside  wires;  and  if  each  of  the  three  neutral 

28 


434         ALTERNATING-CURRENT  PHENOMENA 

wires  is  of  one-half  the  cross-section  of  the  outside  wires,  seven- 
sixty-fourths  or  10.93  per  cent,  of  the  copper. 

Coming  now  to  the  three-phase  system  with  the  potential,  e, 
between  the  lines  as  delta  potential,  if  i  =  the  current  per  line 

or  Y  current,  the  current  from  line  to  line  or  delta  current  =  —=; 


and  since  three  branches  are  used,  the  total  power  is  —   =  =  eii  \/3- 

V3 

Hence  if  the  same  power  has  to  be  transmitted  by  the  three- 
phase  system  as  with  the  single-phase  system,  the  three-phase 

line  current  must  be  i\  =  —  -^;  where  i  =  single-phase  current, 

r  =  single-phase  resistance  per  line,  at  equal  power  and  loss: 
hence  if  r\  =  resistance  of  each  of  the  three  wires,  the  loss  per 

wire  is  i<?r\  =  -5-,  and  the  total  loss  is  iVi,  while  in  the  single- 
o 

phase  system  it  is  2  izr.  Hence,  to  get  the  same  loss,  it  must  be: 
7*1  =  2  r,  that  is,  each  of  the  three  three-phase  lines  has  twice  the 
resistance  —  that  is,  half  the  copper  of  each  of  the  two  single- 
phase  lines;  or  in  other  words,  the  three-phase  system  requires 
three-fourths  as  much  copper  as  the  single-phase  system  of  the 
same  potential. 

Introducing,  however,  a  fourth  or  neutral  wire  into  the  three- 
phase  system,  and  connecting  the  lamps  between  the  neutral 
wire  and  the  three  outside  wires  —  that  is,  in  Y  connection  —  the 
potential  between  the  outside  wires  or  delta  potential  will  be 
=  e  X  \/3,  since  the  Y  potential  =  e,  and  the  potential  of  the 
system  is  raised  thereby  from  eio  e\/3;  that  is,  only  one-third 
as  much  copper  is  required  in  the  outside  wires  as  before  —  that 
is  one-fourth  as  much  copper  as  in  the  single-phase  two-wire  sys- 
tem. Making  the  neutral  of  the  same  cross-section  as  the  out- 
side wires,  requires  one-third  more  copper,  or  ^  =  33.3  per  cent. 
of  the  copper  of  the  single-sphase  sytem;  making  the  neutral 
of  half  cross-section,  requires  one-sixth  more,  or  /^  =  29.17  per 
cent,  of  the  copper  of  the  single-phase  system.  The  system, 
however,  now  is  a  four-  wire  system. 

The  independent  quarter-phase  system  with  four  wires  is 
identical  in  efficiency  to  the  two-wire,  single-phase  system,  since 
it  is  nothing  but  two  independent  single-phase  systems  in  quad- 
rature. 

The  four-wire,  quarter-phase  system  can  be  used  as  two  inde- 


EFFICIENCY  OF  SYSTEMS  435 

pendent  Edison  three-wire  systems  also,  deriving  therefrom  the 
same  saving  by  doubling  the  potential  between  the  outside  wires, 
and  has  in  this  case  the  advantage,  that  by  interlinkage,  the  same 
neutral  wire  can  be  used  for  both  phases,  and  thus  one  of  the 
neutral  wires  saved. 

In  this  case  the  quarter-phase  system  with  common  neutral  of 
full  cross-section  requires  yV  or  31.25  per  cent.,  the  quarter-phase 
system  with  common  neutral  of  one-half  cross-section  requires 
-V  or  28.125  per  cent,  of  the  copper  of  the  two-wire,  single-phase 
system. 

In  this  case,  however,  the  system  is  a  five- wire  system,  and 
as  such  far  inferior  in  copper  efficiency  to  the  five-wire,  single- 
phase  system. 

Coming  now  to  the  quarter-phase  system  with  common  return 
and  potential  e  per  branch,  denoting  the  current  in  the  outside 
wires  by  i%,  the  current  in  the  central  wire  is  2*2\/2;  and  if  the 
same  current  density  is  chosen  for  all  three  wires,  as  the  condition 
of  maximum  efficiency,  and  the  resistance  of  each  outside  wire 

denoted  by  r<z,  the  resistance  of  the  central  wire  =  — ^=.,  and  the 

v  2 

2  izzr2 

loss  of  power  per  outside  wire  is  i^r2)  in  the  central  wire  — j=- 

V2 

=  f»Vr^2 ;  hence  the  total  loss  of  power  is  2  i^r^  +  *22/*2\/2 
=  1*2^2(2  +  \/2).  The  power  transmitted  per  branch  is  i&, 
hence  the  total  power,  2  t"2e.  To  transmit  the  same  power  as  by 

a  single-phase  system  of  power,  ei,  it  must  be  i%  =  75-;  hence  the 

A 

loss,  -  — j—  — .  Since  this  loss  shall  be  the  same  as  the  loss, 
2  izr,  in  the  single-phase  system,  it  must  be  2  r  =  -: r2, 

or  r2  =  — — — ~    Therefore  each  of  the  outside  wires  must  be 

2  -f  \/2 
— g—  -  times  as  large  as  each  single-phase  wire,  the  central 

wire  \/~2  times  larger;  hence  the  copper  required  for  the  quarter- 
phase  system  with  common  return  bears  to  the  copper  required 
for  the  single-phase  system  the  relation, 

2(2+  \/2)       (2+x/2)V"2  3  +  2    /2 

-g-      -  +  -         -g~  -2,  or,  -  -g-      --M,  =  72.9 

per  cent,  of  the  copper  of  the  single-phase  system. 


436         ALTERNATING-CURRENT  PHENOMENA 

Hence  the  quarter-phase  system  with  common  return  saves  2 
per  cent,  more  copper  than  the  three-phase  system,  but  is  inferior 
to  the  single-phase  three-wire  system. 

The  inverted  three-phase  system,  consisting  of  two  e.m.fs.  e  at 
60°  displacement,  and  three  equal  currents  is  in  the  three  lines 
of  equal  resistance  r3,  gives  the  output  2  eis,  that  is,  compared 

with  the  single-phase  system,  iz  =  -5-     The  loss  in  the  three  lines 

Zi 

is  3  ^32r3  =  |  izr$.  Hence,  to  give  the  same  loss,  2  izr,  as  the 
single-phase  system,  it  must  be  r3  =  f  r,  that  is,  each  of  the  three 
wires  must  have  three-eighths  of  the  copper  cross-section  of  the 
wire  in  the  two- wire  single-phase  system;  or  in  other  words,  the 
inverted  three-phase  system  requires  nine-sixteenths  of  the  cop- 
per of  the  two- wire  single-phase  system. 

Thus  if  a  given  power  has  to  be  transmitted  at  a  given  loss, 
and  a  given  minimum  potential,  as  for  instance  110  volts  for 
lighting,  the  amount  of  copper  necessary  is: 

2  WIRES:  Single-phase  system,  100.0 

3  WIRES:  Edison  three- wire  single-phase  system, 

neutral  full  section,  37 . 5 

Edison  three- wire  single-phase  system, 

neutral  half-section,  31.25 

Inverted  three-phase  system,  56 . 25 

Quarter-phase  system  with  common  re- 
turn, 72.9 
Three-phase  system,  75 . 0 

4  WIRES  :  Three-phase,  with  neutral-wire  full  sec- 

tion, 33.3 

Three-phase,    with    neutral-wire    half- 
section,  29.17 
Independent  quarter-phase  system,           100 . 0 
•5  WIRES:  Edison  five- wire,  single-phase  system, 

full  neutral,  15.625 

Edison  five- wire,  single-phase  system, 

half-neutral,  10.93 

Four- wire,    quarter-phase,    with   com- 
mon-neutral full  section,  31.25 
Four-wire,    quarter-phase,    with   com- 
mon-neutral half-section,  28.125 

We  see  herefrom,  that  in  distribution  for  lighting — that  is, 
with  the  same  minimum  potential,  and  with  the  same  number 


EFFICIENCY  OF  SYSTEMS  437 

of  wires  —  the  single-phase  system  is  superior  to  any  polyphase 
system. 

The  continuous-current  system  is  equivalent  in  this  comparison  to 
the  single-phase  alternating-current  system  of  the  same  effective 
potential,  since  the  comparison  is  made  on  the  basis  of  effective 
potential,  and  the  power  depends  upon  the  effective  potential  also. 

296.  Comparison  on  the  Basis  of  Equality  of  the-  Maximum 
Difference  of  Potential  between  any  two  Conductors  of  the  System, 
in  Long-distance  Transmission,  Power  Distribution,  etc. 

Wherever  the  potential  is  so  high  as  to  bring  the  question  of 
the  strain  on  the  insulation  into  consideration,  or  in  other  cases, 
to  approach  the  danger  limit  to  life,  the  proper  comparison  of 
different  systems  is  on  the  basis  of  equality  of  maximum  poten- 
tial in  the  system. 

Hence  in  this  case,  since  the  maximum  potential  is  fixed,  noth- 
ing is  gained  by  three-  or  five-wire,  Edison  systems.  Thus,  such 
systems  do  not  come  into  consideration. 

The  comparison  of  the  three-phase  system  with  the  single- 
phase  system  remains  the  same,  since  the  three-phase  system 
has  the  same  maximum  as  minimum  potential;  that  is: 

The  three-phase  system  requires  three-fourths  of  the  copper 
of  the  single-phase  system  to  transmit  the  same  power  at  the 
same  loss  over  the  same  distance. 

The  four-wire,  quarter-phase  system  requires  the  same  amount 
of  copper  as  the  single-phase  system,  since  it  consists  of  two 
single-phase  systems. 

In  a  quarter-phase  system  with  common  return,  the  potential 
between  the  outside  wires  is  \/~2  times  the  potential  per  branch, 
hence  to  get  the  same  maximum  strain  on  the  insulation  —  that  is, 
the  same  potential,  e,  between  the  outside  wires  as  in  the  single- 

/> 

phase  system  —  the  potential  per  branch  will  be  —  7=,  hence  the 

V  2 

<j 

current  n  =  —  7=.  if  i  equals  the  current  of  the  single-phase  sys- 
V  2 

tern  of  equal  power,  and  it\/2  =  i  will  be  the  current  in  the 
central  wire. 

Hence,  if  r4  =  resistance  per  outside  wire,  —  -  —  =  resistance  of 

V  2 
central  wire,  and  the  total  loss  in  the  syst.em  is 


.     2  ,- 

2  ?42r4  H  --  -7=-  =  24V4(2  +  V  2). 


438         ALTERNATING-CURRENT  PHENOMENA 
Since  in  the  single-phase  system,  the  loss  =  2  22r,  it  is 

4r 


r4  = 


V2 


That- is,  each  of  the  outside  wires  has  to  contain ^ times 

as  much  copper  as  each  of  the  single-phase  wires.  The  central 
wires  have  to  contain 7 \/2  times  as  much  copper;  hence 

the  total  system  contains j '•  -  +  -    -~ — \/2  times  as 

2  i   2  -\/~2 
much  copper  as  each  of  the  single-phase  wires ;  that  is, -r 

£n 

times  the  copper  of  the  single-phase  system. 
Or,  in  other  words, 

A  quarter-phase  system  with  common  return  requires  — 

=  1.457  times  as  much  copper  as  a  single-phase  system  of  the 
same  maximum  potential,  same  power,  and  same  loss. 

Since  the  comparison  is  made  on  the  basis  of  equal  maximum 
potential,  and  the  maximum  potential  of  an  alternating  system  is 
V2  times  that  of  a  continuous-current  circuit  of  equal  effective 
potential,  the  alternating  circuit  of  effective  potential,  e,  com- 
pares with  the  continuous-current  circuit  of  potential  e  \/2, 
which  latter  requires  only  half  the  copper  of  the  alternating 
system. 

This  comparison  of  the  alternating  with  the  continuous-cur- 
rent system  is  not  proper,  however,  since  the  continuous-current 
voltage  may  introduce,  besides  the  electrostatic  strain,  an  elec- 
trolytic strain  on  the  dielectric  which  does  not  exist  in  the  alter- 
nating system,  and  thus  may  make  the  action  of  the  continuous- 
current  voltage  on  the  insulation  more  severe  than  that  of  an 
equal  alternating  voltage.  Besides,  self-induction  having  no 
effect  on  a  steady  current,  continuous-current  circuits  as  a  rule 
have  a  self-induction  far  in  excess  of  any  alternating  circuit. 
During  changes  of  current,  as  make  and  break,  and  changes  of 
load,  especially  rapid  changes,  there  may  consequently  be  gen- 
erated in  these  circuits  e.m.fs.  far  exceeding  their  normal  poten- 
tials. Inversely,  however,  with  alternating  voltages,  dielectric 
hysteresis,  etc.,  may  cause  heating  and  thereby  lower  the 
disruptive  strength.  At  the  voltages  which  came  under  con- 
sideration, the  continuous  current  is  usually  excluded  to  begin 
with. 


EFFICIENCY  OF  SYSTEMS  439 

Thus  we  get: 

If  a  given  power  is  to  be  transmitted  at  a  given  loss,  and  a 
given  maximum  difference  of  potential  in  the  system,  that  is, 
with  the  same  strain  on  the  insulation,  the  amount  of  copper 
required  is: 

2  WIRES:  Single-phase  system,  100.0 

[Continuous-current  system,  50  .  0] 

3  WIRES:  Three-phase  system,  75.0 

Quarter-phase  system,  with  common 

return,  145  .  7 

4  WIRES:  Independent  Quarter-phase  system,  100.0 

Hence  the  quarter-phase  system  with  common  return  is  prac- 
tically excluded  from  long-distance  transmission. 

297.  In  a  different  way  the  same  comparative  results  be- 
tween single-phase,  three-phase,  and  quarter-phase  systems  can 
be  derived  by  resolving  the  systems  into  their  single-phase 
branches. 

The  three-phase  system  of  e.m.f.,  e,  between  the  lines  can  be 
considered  as  consisting  of  three  single-phase  circuits  of  e.m.f., 

p 

,  and  no  return;  the  single-phase  system  of  e.m.f.,  e,  between 


^ 

lines  as  consisting  of  two  single-phase  circuits  of  e.m.f.,  ~  >  and 

A 

no  return.     Thus,  the  relative  amount  of  copper  in  the  two  sys- 
tems being  inversely  proportional  to  the  square  of  e.m.f.,  bears 

/•\/3\  2   /2\  2 
the  relation  (  --  )   :(-)    =3  :4;  that  is,  the  three-phase  sys- 

\    6    I        \6  / 

tern  requires  75  per  cent,  of  the  copper  of  the  single-phase  system. 
The  quarter-phase  system  with  four  equal  wires  requires  the 
same  copper  as  the  single-phase  system,  since  it  consists  of  two 
single-phase  circuits.  Replacing  two  of  the  four  quarter-phase 
wires  by  one  wire  of  the  same  cross-section  as  each  of  the  wires 
replaced  thereby,  the  current  in  this  wire  is  \/2  times  as  large 
as  in  the  other  wires,  hence,  the  loss  is  twice  as  large—  that  is, 
the  same  as  in  the  two  wires  replaced  by  this  common  wire,  or 
the  total  loss  is  not  changed  —  while  25  per  cent,  of  the  copper  is 
saved,  and  the  system  requires  only  75  per  cent,  of  the  copper  of 
the  single-phase  system,  but  produces  \/2  times  as  high  a  poten- 


440         ALTERNATING-CURRENT  PHENOMENA 

tial  between  the  outside  wires.  Hence,  to  give  the  same  maxi- 
mum potential,  the  e.m.fs.  of  the  system  have  to  be  reduced  by 
\/2 ,  that  is,  the  amount  of  copper  doubled,  and  thus  the  quarter- 
phase  system  with  common  return  of  the  same  cross-section  as 
the  outside  wires  requires  150  per  cent,  of  the  copper  of  the  single- 
phase  system.  In  this  case,  however,  the  current  density  in  the 
middle  wire  is  higher,  thus  the  copper  not  used  most  economically, 
and  transferring  a  part  of  the  copper  from  the  outside  wires  to 
the  middle  wire,  to  bring  all  three  wires  to  the  same  current  den- 
sity, reduces  the  loss,  and  thereby  reduces  the  amount  of  copper 
at  a  given  loss,  to  145.7  per  cent,  of  that  of  a  single-phase  system. 

298.  Comparison  on  the  basis  of  equality  of  the  maximum  differ- 
ence of  potential  between  any  conductor  of  the  system  and  the  ground, 
in  long-distance,  three-phase  transmissions  with  grounded  neutral, 
single-phase  systems  with  ground  return,  etc. 

A  system  may  be  grounded  by  grounding  its  neutral  point, 
for  the  purpose  of  maintaining  constant-potential  difference  be- 
tween the  conductors  and  ground,  without  carrying  any  current 
through  the  ground,  or  the  ground  may  be  used  as  return  con- 
ductor. In  either  case  the  system  can  be  considered  as  consist- 
ing of  and  resolved  into  as  many  single-phase  systems  with 
ground  return,  as  there  are  overhead  conductors,  and  with  zero 
resistance  in  the  ground. 

It  immediately  follows  herefrom,  that  the  copper  efficiency  of 
such  a  system  is  the  same  as  that  of  a  single-phase  system  with 
ground  return,  of  the  same  voltage  as  exists  between  conductor 
and  ground  of  the  system  under  consideration.  If  then  all  the 
overhead  conductors  have  the  same  potential  difference  against 
ground,  as  is  the  case  in  a  three-phase  or  quarter-phase  system 
with  grounded  neutral,  a  single-phase  system  with  grounded 
neutral,  or  quarter-phase  system  with  common  ground  return  of 
both  phases,  the  copper  efficiency  is  the  same.  That  is: 

All  grounded  systems,  whether  with  grounded  neutral  or  with 
ground  return,  have  the  same  copper  efficiency,  provided  that 
all  the  overhead  conductors  have  the  same  potential  difference 
against  ground. 

Hence: 

The  three-phase  system  with  grounded  neutral  has-  no  supe- 
riority over  the  single-phase  or  the  quarter-phase  system  with 
grounded  neutral,  in  copper  efficiency.  The  advantage  of  the 
three-phase  system — which  causes  its  practically  universal  use — 


EFFICIENCY  OF  SYSTEMS  441 

over  the  single-phase  system  is  the  greater  usefulness  of  polyphase 
power,  the  advantage  over  the  quarter-phase  system  is  the  use 
of  three  conductors,  against  four  with  the  quarter-phase  system. 

No  saving  in  copper  results  from  the  use  of  the  ground  (of 
zero  resistance)  as  return  circuit,  but  a  single-phase  or  quarter- 
phase  system  with  ground  return,  at  equal  dielectric  strain  on 
the  insulation,  requires  the  same  amount  of  copper  as  a  system 
with  grounded  neutral,  but  has  a  greater  self-induction,  due  to 
the  greater  distance  between  conductor  and  return  conductor  or 
ground,  and  has  the  objection  of  establishing  current  through 
the  ground  and  so  disturbing  neighboring  circuits,  by  electro- 
magnetic and  electrostatic  induction. 

The  apparent  saving  in  copper,  in  the  single-phase  system,  by 
replacing  one  of  the  conductors  by  the  ground  as  return,  there- 
fore is  a  fallacy.  By  doing  so,  the  potential  difference  of  the  other 
conductors  against  ground  becomes  twice  what  it  would  be  with 
two  conductors  and  grounded  neutral,  and  at  the  same  potential 
difference  between  conductors.  That  is,  the  single-phase  system 
with  ground  return  requires  the  same  insulation  as  a  single-phase 
system  with  grounded  neutral,  of  twice  the  voltage,  and  then  re- 
quires the  same  copper.  A  saving  results  only  in  the  number  of 
insulators  required,  etc.  Only  where  the  amount  of  power  is  so 
small  that  mechanical  strength,  and  not  power  loss,  determines 
the  size  of  the  conductor,  a  saving  results  by  replacing  one  of  the 
conductors  by  the  ground. 

The  high-tension,  direct-current  system,  whether  insulated,  or 
with  grounded  neutral,  or  with  ground  return,  appears  equal  in 
copper  efficiency  to  a  single-phase  system  of  the  same  character 
(insulated,  or  with  grounded  neutral,  or  with  ground  return)  and 
of  the  same  effective  voltage,  that  is,  with  a  sine  wave  of  a  maxi- 
mum voltage  V2  times  that  of  the  direct  current.  Due  to  the 
different  character  of  unidirectional  electric  stress  of  the  direct- 
current  system,  from  the  alternating  stress,  a  general  comparison 
of  the  system  by  a  numerical  factor  appears  hardly  feasible.  It 
is,  however,  claimed  that  usually  the  insulation  stress  with  per- 
fectly uniform  continuous  voltage  is  less  than  that  of  an  alter- 
nating voltage  of  the  same  maximum  value,  so  that  continuous- 
current  high-voltage  transmission  would  offer  advantages,  if  it 
were  not  for  the  difficulty  of  generating  and  utilizing  very  high 
continuous  voltages,  which  with  alternating  voltages  is  overcome 
by  the  interposition  of  the  stationary  transformer. 


CHAPTER  XXXIV 
METERING  OF  POLYPHASE  CIRCUIT 

299.  The  power  of  a  polyphase  system  or  circuit  is  the  sum  of 
the  powers  of  all  the  individual  branch  circuits,  and  the  sum  of 
the  wattmeter  readings  of  all  the  branch  circuits  thus  gives  the 
total  power. 

Let,  then,  in  a  general  polyphase  system,  e\,  62,  63  .  .  .  en  = 
potentials  at  the  n  terminals  or  supply  wires  of  the  r?-phase 
system. 

These  may  be  represented  topographically  by  points  in  a  plane, 
as  shown  in  Fig.  218. 


FIG.  218. 
The  voltage  between  any  two  terminals  e*  and  ek  then  is: 

eik  =  e<  —  ek  (1) 

And  this  voltage,  in  any  circuit  connected  between  these  two 
terminals,  produces  a  current,  iik,  as  the  current,  which  flows  from 
6i  to  6k  through  this  circuit. 

As  there  are  ~ pairs  of  terminals  €i  and  ek,  there  are 

z 

existing  in  a  general  n-phase  system  — -= different  phases, 

A 

and  there  may  thus  be -t    —  different  circuits,  or  rather  sets 

z 

442 


METERING  OF  POLYPHASE  CIRCUIT  443 

of  circuits  —  since  a  number  of  circuits  may  and  usually  are  con- 
nected between  the  n  terminals. 

Consider  one  of  these  numerous  circuits  of  the  general  n-phase 
system,  that  of  the  current  {•&  passing  from  d  to  ek.  The  power 
of  this  circuit  is: 

Pik  =  [ei  -  ek,  iik]  (2) 

where  the  brackets  denote  the  effective  power,  as  discussed  in 
Chapter  XVI. 

Choosing  any  point  ex,  which  may  be  one  of  the  terminals,  or 
the  neutral  point  of  the  system,  if  such  exists,  or  any  other  point. 
Then  the  voltage  et-  —  ek  can  be  resolved  by  the  parallelogram 
(Fig.  218)  into  the  voltages:  e^  —  ex  and  ex  —  e^, 
that  is: 

ei  —  ek  =  6i  —  ex  +  ex  —  ek  (3) 

hence,  substituted  into  (2)  : 

Pik  =  [(ei  -  xe)  +  (ex  -  ek),  iik] 

=  [ei  -  ex,  iik]  +  [ex  -  ek,  iik]  (4) 

It  is,  however: 

[ex  —  ek,  iik]  =  [ek  —  ex,  iki]  (5) 

where  iki  is  the  current  flowing  from  ek  to  ei}  that  is,  the  same  cur- 
rent as  iikt  only  considered  in  the  reverse  direction. 
Thus  it  is,  substituting  (5)  into  (4)  : 

Pik  =  [ei  —  ex,  iik]  +  [ek  —  ex,  iki]  (6) 

That  is,  the  power  of  any  branch  circuit  between  two  terminals, 
6i  and  ek)  is  the  product  of  the  powers  giving  by  the  two  potential 
differences  e»  —  ex  and  ek  —  ex,  of  any  arbitrarily  chosen  point 
ex,  with  the  current  flowing  into  this  branch  circuit  from  the  two 
terminals,  e»  and  ek,  that  is,  iik  and  4i,  respectively. 

300.  The  total  power  of  the  n-phase  system,  as  the  sum  of  the 
powers  of  all  the  branch  circuits,  then  is: 


=  S.'         [e{  -  ex,  iik]  (7) 

i      i 

where  the  double  summation  sign  indicates  that  the  summation 
is  to  be  carried  out  for  all  values  of  k,  from  1  to  n,  and  for  all 
values  of  i,  from  1  to  n. 


444         ALTERNATING-CURRENT  PHENOMENA 

As  the  term  e»  —  ex  in  (7)  does  not  contain  the  index  k,  it  is 
the  same  for  all  values  of  k,  thus  can  be  taken  out  from  the  second 
summation  sign,  that  is: 

P  =  S«'  L  -  ef,  2*  i  J  (8) 

However : 

n 

2*  i ik  is  the  sum  of  all  the  currents,  flowing  from  the  termi- 

i 

nal  6i  to  all  the  other  terminals  eh  (k  =  1,  2   .    .   n),  that  is,  it  is 
the  total  current  issuing  from  the  terminal  et,  or: 

it  =  2*  iit  (9) 

i 

and,  substituting  this  in  9,  gives  as  the  total  power  of  the  n-phase 
system : 

P  =  Si  [a  -  ex,  i,]  (10) 

i 

That  is: 

"The  total  power  of  a  general  n-phase  system,  is  the  sum  of  the 
n  powers,  given  by  the  n  currents  ii,  which  issue  from  the  n 
terminals  e^  with  the  n  potential  differences  of  these  terminals  e. 
against  any  arbitrarily  chosen  point  ex." 

"The  total  power  of  the  system,  no  matter  how  many  branch  cir- 
cuits it  contains,  thus  is  measured  by  n  wattmeters. 

Choosing  as  the  point,  ex  one  of  the  n-phase  circuit  terminals, 
that  is  one  of  the  phase  potentials  (for  instance,  the  neutral 
potential  of  the  system,  where  such  exists),  as  en,  the  number  of 
terms  in  (10)  reduces  by  one: 

P  =n2i[ei-  e^ii]  (11) 

i 

That  is: 

"The  total  power  of  a  general  n-phase  s-ystem  is  measured  by 
n  —  I  wattmeters,  connected  between  one  terminal  en  and  the  n  —  1 
other  terminals  e\" 

Thus  for  instance,  a  five-wire,  four-phase  system  (Fig.  195), 

5X4 
in  which  — ^ —  =  10  different  sets  of  circuits  are  possible,   is 

metered  by  5  —  1  =  4  meters. 

A  four-wire,  three-phase  system  is  metered  by  3  meters. 
A  three-wire,  three-phase  system  is  metered  by  2  meters. 


METERING  OF  POLYPHASE  CIRCUIT 


445 


301.  In  a  three-phase  system  with  ungrounded  neutral,  that 
is,  a  three-wire,  three-phase  system,  the  common  method  of 
measuring  the  total  power  thus  is,  by  (11),  as  shown  in  Fig.  219. 

Often  the  two  meters  of  Fig.  219  are  arranged  in  one  structure. 


(MT 

m 

b 

HI 
f 

~\Ml 

nn 

£ 

UU  — 

FIG.  219. 


Thus,  if  Fig.  220  denotes  a  general  three-wire,  three-phase  sys- 
tem, with  the  voltages  and  currents  in  the  three  phases: 

Ei  E2  E3  and  Ii  72  7 


2,    3 


counting  voltages  and  currents  in  the  direction  indicated  by  the 
arrows  in  Fig.  220. 


FIG.  220. 

The  voltages  may  be  unequal  in  sizes  and  under  unequal 
angles,  by  a  distortion  of  the  three-phase  triangle,  but  it  must  be : 

Ei  +  E2  +  E,  =  0  (12) 

in  a  closed  triangle. 

Connecting  then  the  current  coils  of  the  two  wattmeters  into 
the  lines  a  and  6,  and  the  voltage  coils  between  a  respectively  b, 
and  c,  the  two  wattmeter  readings  are: 


and: 


-  Ei,  72  -  7i]  = 
[#3,  Is  -  h]  = 


72] 


(13) 


446 


AL  TERN  A  TING-C  URREN  T  PHENOMENA 


and  their  sum  is: 

P  =  [Ei,  Ii]  ~ 
=  [Elt  h]  - 

and  since  by  (12)  : 


/  J  -  [E»  /,]  +  [Ei,  /,] 
+  0,,  I2]  +  [E3,  /  j 


(14) 


it  is: 


P  = 


3,  /a] 


that  is,  the  total  power  of  the  three-phase  system  is  the  sum  of 
the  individual  powers  of  the  three  branch  circuits. 

302.  In  the  standard  polyphase  wattmeter  connection  of  the 
three-wire,  three-phase  system,  Fig.  219,  the  voltage  coils  are 
out  of  phase  with  the  current  coils  at  non-inductive  load,  the  one 
lagging,  the  other  leading  by  30°.  Therefore,  even  in  a  balanced 


FIG.  221. 

system,  if  the  current  lags,  the  two  wattmeter  coils  do  not  read 
alike,  as  the  voltmeter  coil  in  the  one  lags  by  the  angle  of  lag  of 
the  current  plus  30°,  and  in  the  other  by  the  angle  of  lag  minus 
30°.  At  60°  angle  of  lag,  the  voltage  coil  of  the  former  lags 
60  +  30  =  90°,  and  the  reading  becomes  zero,  and  at  more  than 
60°  lag,  the  one  meter  reads  negative,  but  the  algebraic  sum  of  the 
two  meter  readings  still  remains  the  total  power  of  the  circuit, 
the  one  meter  reading  more  than  the  total  power,  while  the  other 
meter  reads  negative. 

In  a  balanced,  or  nearly  balanced  three-wire,  three-phase  sys- 
tem, instead  of  connecting  the  potential  coils  from  a  and  6  to  c, 
Figs.  219  and  220,  they  are  often  connected  from  a  to  b.  This 
interchanges  the  lagging  and  the  leading  coil,  but  on  balanced 
loads  leaves  the  same  total.  In  this  case,  one  voltage  coil  only 
may  be  used,  acted  upon  by  two  current  coils.  That  is,  a  single- 
phase  wattmeter  is  constructed,  similar  to  the  Edison  three- wire 


METERING  OF  POLYPHASE  CIRCUIT 


447 


meter,  with  one  current  coil  in  the  one,  the  other  current  coil  in 
the  other  line,  and  the  voltage  coil  connected  between  these  two 
lines,  as  shown  in  Fig.  221. 

If  there  is  considerable  unbalancing,  this   latter   connection 
gives  considerable  error,  and  the  double  meter  has  to  be  used. 


^MT 

nn 

A 

UU 

~ULM 

(V) 

(• 

HI 

£ 

FIG.  222. 

In  a  four-wire,  three-phase  system,  the  connection  of  the  two 
meters  obviously  becomes  wrong,  if  current  flows  in  the  neutral, 
and  three  meters  must  be  used. 

Most  conveniently  these  are  arranged  with  the  three  current 
coils  in  the  three  lines,  and  the  voltage  coils  between  these  lines 
and  the  neutral,  as  shown  in  Fig.  222. 


CHAPTER  XXXV 
BALANCED  SYMMETRICAL  POLYPHASE  SYSTEMS 

303.  In  most  applications  of  polyphase  systems  the  system  is  a 
balanced  symmetrical  system,  or  as  nearly  balanced  as  possible. 
That  is,  it  consists  of  n  equal  e.m.fs.  displaced  in  phase  from  each 

other  by  —  period,  and  producing  equal  currents  of  equal  phase 

displacement  against  their  e.m.fs.  In  such  systems,  each  e.m.f. 
and  its  current  can  be  considered  separately  as  constituting  a 
single-phase  system,  that  is,  the  polyphase  system  can  be  resolved 
into  n  equal  single-phase  systems,  each  of  which  consists  of  one 
conductor  of  the  polyphase  system,  with  zero  impedance  as  return 
circuit.  Hereby  the  investigation  of  the  polyphase  system 
resolves  itself  into  that  of  its  constituent  single-phase  system. 

So,  for  instance,  the  polyphase  system  shown  in  Fig.  208,  at 
balanced  load,  can  be  considered  as  consisting  of  the  equal  single- 
phase  systems  :0  —  1;0  —  2;  0  —  3;  .  .  .  0  —  r&,  each  of 
which  consists  of  one  conductor,  1,  2,  3,  .  .  .  n,  and  the  return 
conductor,  0.  Since  the  sum  of  all  the  currents  equals  0,  there  is 
no  current  in  conductor  0,  that  is,  no  voltage  is  consumed  in  this 
conductor;  this  is  equivalent  to  assuming  this  conductor  as  of 
zero  impedance.  This  common  return  conductor,  0,  since  it 
carries  no  current,  can  be  omitted,  as  is  usually  the  case.  With 
star  connection  of  an  apparatus  into  a  polyphase  system,  as  in 
Fig.  200,  the  impedance  of  the  equivalent  single-phase  system  is 
the  impedance  of  one  conductor  or  circuit;  if,  however,  the  appa- 
ratus is  ring  connected,  as  shown  diagrammatically  in  Fig.  201, 
the  impedance  of  the  ring-connected  part  of  the  circuit  has  to 
be  reduced  to  star  connection,  in  the  usual  manner  of  reducing 
a  circuit  to  another  circuit  of  different  voltage,  by  the  ratio 

ring  voltage 

s»      —••      . • 

star  voltage' 
or,  as  these  voltages  are  usually  called  in  a  three-phase  system, 

_  delta  voltage 
Y  voltage 
448 


BALANCED  SYMMETRICAL  POLYPHASE  SYSTEMS  449 

That  is,  all  ring  voltages  are  divided,  all  ring  currents  multiplied 
with  c;  all  ring  impedances  are  divided,  all  ring  admittances 
multiplied  with  the  square  of  the  ratio,  c2. 

For  instance,  if  in  a  three-phase  induction  motor  with  delta- 
connected  circuits,  the  impedance  of  each  circuit  is 

Z    =  r  +  jx, 

and  the  voltage  impressed  upon  the  circuit  terminals  E,  and  the 
motor  is  supplied  over  a  line  of  impedance,  per  line  wire, 

ZQ   =  r0  +  JX0, 

the  motor  impedance,  reduced  to  star  connection,  or  Y  impe- 
dance, is 

,         r  4-  jx       1  t         .  A 
Z'  =  r  —  ^—  =  3  (r+jx)-t 

and  the  impressed  voltage,  reduced  to  Y  circuit, 

E 

w  - 

?  -  vf     ,     I 

and  the  total  impedance  of  the  equivalent  single-phase  circuit  is 
therefore 

Z0  +  Z'  =  (r0  +  jxo)  +      (r  +  jx). 


Inversely,  however,  where  this  appears  more  convenient,  all 
quantities  may  be  reduced  to  ring  or  delta  connection,  or  one  of 
the  ring  connections  considered  as  equivalent  single-phase  circuit, 
of  impedance 

Z  +  c2Z0  =  (r  +  jx)  +  3(r0  +  jx0). 

Since  the  line  impedances,  line  currents  and  the  voltages  con- 
sumed in  /the  lines  of  a  polyphase  system  are  star,  or  (in  a  three- 
phase  system)  Y  quantities,  it  usually  is  more  convenient  to 
reduce  all  quantities  to  Y  connection,  and  use  one  of  the  F-cir- 
cuits  as  the  equivalent  single-phase  circuit. 

304.  As  an  example  may  be  considered  the  calculation  of  a 
long-distance  transmission  line,  delivering  10,000  kw.,  three-phase 
power  at  60  cycles,  80,000  volts  and  90  per  cent,  power-factor  at 
100  miles  from  the  generating  station,  with  approximately  10  per 
cent,  loss  of  power  in  the  transmission  line,  and  with  the  line 
conductors  arranged  in  a  triangle  6  ft.  distant  from  each  other. 

29 


450         ALTERNATING-CURRENT  PHENOMENA 

10,000  kw.  total  power  delivered  gives  3,333  kw.  per  line  or 
single-phase  branch  (F  power). 

3,333  kw.  at  90  per  cent,  power-factor  gives  3,700  kv.-amp. 

80,000  volts  between  the  lines  gives  80,000  -^  \/3  =  46,100 
volts  from  line  to  neutral,  or  per  single-phase  circuit. 

3,700  kv.-amp.  per  circuit,  at  46,100  volts,  gives  80  amp.  per 
line. 

10  per  cent,  loss  gives  333  kw.  loss  per  line,  and  at  80  amp.,  this 
gives  a  resistance  per  line, 

333,000  -f-  802  =  52  ohms, 

or,  0.52  ohms  per  mile. 

The  nearest  standard  size  of  wire  is  No.  0  B.  &  S.,  which  has  a 
resistance  of  0.52  ohms,  and  a  weight  of  1680  Ib.  per  mile. 

Choosing  this  size  of  wire  so  requires  for  the  300  miles  of  line 
conductor,  300  X  1680  =  500,000  Ib.  of  copper. 

At  0.52  ohms  per  mile,  the  resistance  per  transmission  line  or 
circuit  of  100  miles  length  is, 

r  =  52  ohms. 

The  inductance  of  wire  No.  0,  with  d  =  0.325  in.  diameter,  and 
6  ft.  =  72  in.  distance  from  the  return  conductor,  is  calculated 
from  the  formula  of  line  inductance1  as,  2.3  mil-henrys  per  mile; 
hence,  per  circuit, 

L  =  0.23  henry, 


and  herefrom  the  reactance, 


27T/L 

88  ohms. 


The  capacity  of  the  transmission  line  may  be  calculated  directly, 
or  more  conveniently  it  may  be  derived  from  the  inductance.  If 
C  is  the  capacity  of  the  circuit,  of  which  the  inductance  is  L,  then 


4v£c 

is  the  fundamental  frequency  of  oscillation,  or  natural  period, 
that  is,  the  frequency  which  makes  the  length,  I,  of  the  line  a 
quarter- wave  length. 

Since  the  velocity  of  propagation  of  the  electric  field  is  the  ve- 

1  "Theoretical  Elements  of  Electrical  Engineering." 


BALANCED  SYMMETRICAL  POLYPHASE  SYSTEMS  451 

locity  of  light,  v,  with  a  wave-length,  4  I,  the  number  of  waves  per 
second,  or  frequency  of  oscillation  of  the  line,  is 

fi  =  n 

and  herefrom  then  follows: 

L          1      * 

i  ""VLC' 

hence,  for 

I  =  100  miles, 

v  =  186,000  miles  per  second, 
L  =  0.23  henry, 
C  =  1.26  mf. 

and  the  capacity  susceptance, 

b  =  2  TT/C  =  475  X  10-6. 

Representing,  as  approximation,  the  line  capacity  by  a  con- 
denser shunted  across  the  middle  of  the  line 
We  have,  impedance  of  half  the  line, 

Z  =  2  -h  j  g"  =  26  +  44  j  ohms. 

Choosing  the  voltage  at  the  receiving  end  as  zero  vector, 
e  =  46,100  volts, 

at  90  per  cent,  power-factor  and  therefore  43.6  per  cent,  induc- 
tance factor,  the  current  is  represented  by 

I  =  80  (0.9  -  0.436  j)  =72-35  j. 

1Or,   if  fj.  =  permeability,  K  =  dielectric   constant  of  the  medium  sur- 
rounding the  conductor,  it  is 

v 


hence, 

f  =  Mi? 

or, 


C  = 


452         ALTERNATING-CURRENT  PHENOMENA 

This  gives: 

Voltage  at  receiver  circuit,  e  =  46,100  volts; 

current  in  receiver  circuit,  /  =  72  —  35  j  amp.  ; 

impedance  voltage  of  half  the  line,  ZI  =  3410  +  2260  j  volts. 

Hence,  the  condenser  voltage,  Ei  =  e  +  ZI  =  49,510  +  2260  j 
volts  ; 

and  the  condenser  current,  +  jbEi  =  —  1.1  -f  23.8  j  amp.; 

hence,  the  total,  or  generator  current,  70  =  /  +  jbEi  =  70.9  — 
11.2  j  amp. 

The  impedance  voltage  of  the  other  half  of  the  line,  Z/0  = 
2330  -  2830  j  volts; 

hence,  the  generator  voltage,  E0  =  EI  +  ZIQ  =  51,840  +  5090 
j  volts; 
and  the  phase  angle  of  the  generator  current, 

tan  0i=  ^  =  0.158;   0i  =  9.0° 

The  phase  angle  of  the  generator  voltage, 

5090 
tan  02  =  -  =  -  0.098;  02  =  -  5.6°; 


the  lag  of  the  generator  current,  00  =  0i  —  02  =  14.6°; 

hence  the  power-factor  at  the  generator,  cos  00  =  96.7  per  cent. 

And  the  power  output,  3  [/,  e]1  =  10,000  kw.; 

the  power  input,  3  [/0,  ^o]1  =  11,190  kw.; 

the  efficiency  =  89.35  per  cent.; 

the  volt-ampere  output,  3  ie  =  11,110  kv.-amp.; 

the  volt-ampere  input,  3  i^  =  11,220  kv.-amp.; 

ratio:   =  99.02  per  cent. 
And  the  absolute  values  are: 

receiver  current,  i  =  80  amp.  ; 

receiver  voltage,  e  =  46,100  X  \/3  =  80,000  volts; 

generator  current,  iQ  =  71.8  amp.; 

generator  voltage,  eQ  =  52,100  X  \/3  =  90,000  volts; 

voltage  drop  in  line,  =  11.1  per  cent. 

305.  Balanced  polyphase  systems  thus  can  be  calculated  as 
single-phase  systems,  and  this  has  been  done  in  many  preceding 
chapters,  as  in  those  on  the  induction  machines,  synchronous 
machines,  etc.,  that  is,  apparatus  which  is  usually  operated  on 
polyphase  circuits. 


BALANCED  SYMMETRICAL  POLYPHASE  SYSTEMS  453 

Only  in  dealing  with  those  phenomena  which  are  resultants  of 
all  the  phases  of  the  polyphase  system,  in  the  resolution  of  the 
polyphase  system  into  its  constituent  single-phase  systems  the 
effective  value  of  the  constant  has  to  be  used,  which  corresponds 
to  the  resultant  effect.  This,  for  instance,  is  the  case  in  calcu- 
lating the  magnetic  field  of  the  induction  machine  —  which  is 
energized  by  the  combination  of  all  phases  —  or  the  armature 
reaction  of  synchronous  machines,  etc. 

For  instance,  in  the  induction  machine,  from  the  generated 
e.m.f.,  e  —  in  Chapter  XVIII  —  the  magnetic  flux  of  the  machine 
is  calculated,  and  from  the  magnetic  flux  and  the  dimensions  of 
the  magnetic  circuit:  length  and  section  of  air-gap,  and  length  and 
section  of  the  iron  part,  follows  the  ampere-turns  excitation,  that 
is,  the  ampere  turns,  FQ,  required  to  produce  the  magnetic  flux. 

The  resultant  m.m.f.  of  m  equal  magnetizing  coils  displaced 

in  position  by  :L  cycle,  energized  by  m  equal  currents  of  an 
m-phase  system,  is  given  by  §271  as 

nml 

r  o=  —  7= 

\/2 

where 

I   =  current  per  phase,  or  per  magnetizing  coil, 
n  =  number  of  turns  per  coil, 
m  =  number  of  phases. 

The  exciting  current  per  phase  required  to  produce  the  resulting 
m.m.f.,  FQ,  therefore,  is 

T 
1  = 

nm 

hence,  for  a  three-phase  system, 


and  for  a  quarter-phase  system,  with  two  coils  in  quadrature, 


In  the  investigation  of  the  armature  reaction  of  synchronous 
machines,  Chapter  XXII,  the  armature  reaction  of  an  m-phase 
machine  is,  by  §271, 

F  = 


454         ALTERNATING-CURRENT  PHENOMENA 

where 

m  —  number  of  phases, 

no  =  number  of  turns  per  phase,  effective,  that  is,  allow- 
ing for  the  spread  of  turns  over  an  arc  of  the  periph- 
ery in  machines  of  distributed  winding, 

I    =  current  per  phase, 

and  when,  in  Chapter  XX,  the  armature  reaction  is  given  by  nl, 
the  number  of  effective  turns,  n,  is,  accordingly,  for  a  polyphase 
alternator, 

m 


hence,  in  a  three-phase  machine, 

n  =  —~  =  1.5  nQ 
V2 

in  a  quadrature-phase  machine, 

n  =  UQ  -\/2- 

306.  When  replacing  a  balanced  symmetrical  polyphase  system 
by  its  constituent  single-phase  systems,  it  must  be  considered, 
that  the  constants  of  the  constituent  single-phase  circuit  may  not 
be  the  same  which  this  circuit  would  have  as  independent  single- 
phase  circuit. 

If  the  branches  of  the  polyphase  circuit,  which  constitute 
the  equivalent  single-phase  circuits,  are  electrically  or  magnetic- 
ally interlinked,  the  constants,  as  admittance,  impedance,  etc., 
of  the  equivalent  single-phase  circuit  often  are  different  from 
those  of  the  same  circuit  on  single-phase  supply,  and  the  poly- 
phase values  then  must  be  used  in  the  equivalent  single-phase 
circuits  which  replace  the  polyphase  system. 

This  is  the  case  in  induction  machines,  in  the  armatures  of 
synchronous  machines,  etc.,  where  the  phases  are  in  mutual  in- 
duction with  each  other. 

Let,  in  a  star  or  Y-connected  three-phase  induction  motor: 

Y  =  g  -  jb 

be  the  exciting  admittance  and  e  the  impressed  voltage  per  three- 
phase  Y  circuit  or  constituent  single-phase  circuit. 


BALANCED  SYMMETRICAL  POLYPHASE  SYSTEMS  455 
The  exciting  current  per  circuit  then  is: 

I  =  eY 

or,  absolute: 

i  =  ey 

if  n  =  number  of  turns  per  circuit, 

/  =  ni  =  effective  value  of  the  m.m.f.  per  phase,  and 

F=  1.5  X  \/2  ni  =  resultant  m.m.f.  of  all  three  phases. 

F  then  produces  in  the  magnetic  circuit  the  flux  <f>,  which  con- 
sumes the  impressed  voltage  e. 

Assuming  now,  that  instead  of  impressing  three  three-phase 
voltage  e  on  the  three  constituent  single-phase  circuits  of  the 
motor,  we  impress  only  a  single-phase  voltage  e  on  one  of  the 
three  circuits. 

,  The  current  in  this  circuit  then  must  produce  the  same  flux  $, 
and  have  the  same  maximum  m.m.f.  F,  as  was  given  by  the  re- 
sultant of  all  three  phases. 

With  n  turns,  that  means,  the  current  ii  under  the  single- 
phase  e.m.f.  e  is  given  by: 

F  =  \/2  nil 

and  since  we  had,  under  the  same  voltage  e  and  flux  <£,  three- 
phase  : 

F  =  1.5  V2W 
it  follows: 

ii  =  1.5  i 

That  is,  with  a  single-phase  voltage,  e,  the  current,  ii,  and  thus 
the  admittance,  YI,  of  the  circuit,  is  1.5  times  the  current,  i,  and 
thus  the  admittance,  F,  which  is  produced  in  the  same  circuit 
by  the  three-phase  voltage: 

Fi  =  1.5  T 
or: 

Y  =  %  F! 
That  is: 

If  we  measure  the  admittance  of  one  of  the  motor  circuits  by 
single-phase  supply  voltage,  this  is  not  the  admittance  of  this 
circuit  as  constituent  single-phase  circuit  of  the  three-phase 
motor,  but 

The  admittance  of  the  constituent  or  equivalent  single-phase 


456          ALTERNATING-CURRENT  PHENOMENA 

circuit  of  a  three-phase  induction  motor  is  two-thirds  of  the  ad- 
mittance of  this  same  circuit  as  independent  single-phase  circuit. 

We  can  look  at  this  in  a  different  way: 

As  the  three-phase  circuits  combine  to  a  resultant  which  is 
1.5  times  the  m.m.f.  of  each  circuit,  each  circuit  requires  only 
two-thirds  of  the  m.m.f.,  and  thus  two-thirds  of  the  exciting 
admittance,  as  equivalent  single-phase  circuit  of  a  three-phase 
motor,  which  it  would  require,  if  as  independent  single-phase 
circuit  it  had  to  produce  the  entire  m.m.f. 

307.  The  same  applies  to  the  self-inductive  reactance:  as  the 
self-inductive  or  leakage  flux,  which  consumes  the  reactance 
voltage,  is  produced  by  the  resultant  of  the  currents  of  all  three 
phases,  and  this  resultant  is  1.5  times  the  maximum  of  one  phase, 
each  phase  produces  only  two-thirds,  that  is,  the  impedance 
current  of  each  phase  of  the  motor  on  three-phase  voltage  supply 
is  only  two-thirds  that  of  the  same  circuit  at  the  same  voltage 
of  single-phase  supply,  and  the  impedance  thus  is  %  =  1.5 
times. 

That  is: 

The  effective  admittance  of  the  equivalent  or  constituent  sin- 
gle-phase circuit  of  a  three-phase  induction  machine  is  two-thirds 
of  the  admittance,  and  the  effective  impedance  is  1.5  times  the 
impedance  of  this  circuit  as  independent  single-phase  circuit. 

The  same  applies  to  synchronous  machines: 

The  three-phase  synchronous  reactance  per  armature  circuit, 
that  is,  the  synchronous  reactance  of  this  armature  circuit  as 
equivalent  single-phase  circuit  of  the  three-phase  system,  is 
1.5  times  the  single-phase  synchronous  reactance  of  the  same 
armature  circuit,  that  is,  synchronous  reactance  of  this  circuit 
as  single-phase  machine. 

In  dealing  with  the  constituent  single-phase  circuits  of  a  three- 
phase  system,  the  proper  "  three-phase "  values  of  the  constants 
of  the  equivalent  circuit  must  be  used. 


CHAPTER  XXXVI 
THREE-PHASE  SYSTEM 

308.  With  equal  load  of  the  same  phase  displacement  in  all 
three  branches,  the  symmetrical  three-phase  system  offers  no 
special  features  over  those  of  three  equally  loaded  single-phase 
systems,  and  can  be  treated  as  such;  since  the  mutual  reactions 
between  the  three  phases  balance  at  equal  distribution  of  load, 
that  is,  since  each  phase  is  acted  upon  by  the  preceding  phase  in 
an  equal  but  opposite  manner  as  by  the  following  phase. 

With  unequal  distribution  of  load  between  the  different 
branches,  the  voltages  and  phase  differences  become  more  or  less 
unequal.  These  unbalancing  effects  are  obviously  maximum  if 
some  of  the  phases  are  fully  loaded,  others  unloaded. 

Let  E  =  e.m.f.  between  branches  1  and  2  of  a  three-phaser. 
Then 

e  E  =  e.m.f.  between  2  and  3, 
ezE  =  e.m.f.  between  3  and  1; 

where  e  =  \/H  =  —  . 

2 

Let 

Zi,  Z2,  Zz  =  impedances  of  the  lines  issuing  from  generator 

terminals  1,  2,  3, 

and  FI,  F2,  F3  =  admittance^    of   the    consumer   circuits    con- 
nected between  lines  2  and  3,  3  and  1,  1  and  2. 
If  then, 

Ii,  72,  Is,  are  the  currents  issuing  from  the  generator  termi- 
nals into  the  lines,  it  is, 

II    +  /2    +    h    =    0.  (1) 

If,  7'i,  7'2,  7'3  =  currents  through  the  admittances,  FI,  F2,  F3, 
from  2  to  3,  3  to  1,  1  to  2,  it  is, 

Ii  =  73'  -  7'2,  or,  /!  +  7'2  -  7'3  =  0 
72  =  K  -  K,  or,  72  +  7'3  -  I\  =  0  (2) 

h  =  7'2  -  K,  or,  73  +  7'i  -  7'2  =  0 
457 


458         ALTERNATING-CURRENT  PHENOMENA 


These    three    equations    (2)    added,    give   (1)    as    dependent 
equation. 

At  the  ends  of  the  lines  1,  2,  3,  it  is: 


fj   1    =    E\    —    Ziy.1 2   ~f" 

E'z  =  Ez  —  Zili  -\- 
the  differences  of  potential,  and : 

I' i  =  E\Y 
I>2  =  E'2Y 
Tf  =  l?f  V 


(3) 


(4) 


the  currents  in  the  receiver  circuits. 

These  nine  equations  (2),  (3),  (4),  determine  the  nine  quan- 
tities: 1 1,  72,  /3,  I/,  72',  //,  Ei',  E*',  E,'. 

Equations  (4)  substituted  in  (2)  give: 

777>/     V"  7?'     V 

I    =    E   3/3    —    E  2i   2 

/W   V  77"    V  /'K^ 

2    =    E   ill    —   Hi   3^3  (5) 

These  equations  (5)  substituted  in  (3),  and  transposed,  give: 
since       E\  =  e  E 


E3  =  E 


as  e.m.fs.  at  the  generator  terminals. 


e  E  -  E\(l 
e*E  -  E'2(l 


E  ' 


=  0 

3    =    0 

i  =  0 


(6) 


as  three  linear  equations  with  the  three  quantities,  Er\t  E"2,  #'3. 


THREE-PHASE  SYSTEM 


459 


=  I  FA, 

FA, 

we  have: 


hence, 


Substituting  the  abbreviations: 

-(1  +  FA  +  FA),  FA,  F3Z2 

K  =    FA,  -  (1  +  FA  +  FA), 

A,  F2Z!,  -  (1  +  FA  +  FA) 

,    FA,  F3Z2 

2,   -  (1  +  F2Z3  +  F2ZO,       F3Z! 
I,   Y*Zlt 


-  (1  +  FgZi  4- 

F3Z2 


(7) 


1,         -  (1 


FA),  F2Z3, 

-  (1  +  FA  +  F2ZO, 


+  FA) 

e 


K 


. 


EK 


(8) 


I'f- 


(9) 


J,  = 


(10) 


E\  +  E'2  +  E' 3  =  0  I 

/I       +   /2       +   /8       «   0    j 


(ID 


460         ALTERNATING-CURRENT  PHENOMENA 
309.  SPECIAL  CASES. 

A.    Balanced  System 

FV  V  V 

i  =  1 2  =  1 3  =  y 

Substituting  this  in  (6),  and  transposing: 


#!  = 

E2  = 

E,= 
//    . 

E 

eE 

Y 

eE 

1  +3FZ 

f  2  "  1  +  3  YZ 
E 

1  +  3  YZ 

e2  (e  -  1)  EY 

i  - 

/'„  = 

T/ 

1  +  3YZ 

1  +  3FZ 

r          (6-l)EF 

1  +  3FZ 
EY 

1  +  3FZ 
€(e-l)£T 

JL  3 

1  +  3FZ 

1  -f  3FZ 

(12) 


The  equations  of  the  symmetrical  balanced  three-phase  system. 

B.     One  Circuit  Loaded,  Two  Unloaded 
F!=  F2  =  0,  F3  =  F, 

£j\    =    ^2    =    ^3    ==    ^» 


Substituted  in  equations  (6) : 


unloaded  branches. 


e  E  -  E'i  -H  J^3'FZ  =  0 

> 
E  -  E's  (1  +  2  FZ)  =  0,  loaded  branch. 


hence : 


1  +2FZ 


YZ} 


1+2YZ 


f 

1+  2FZ 


unloaded; 


loaded ; 


all  three 
e.m.fs. 
unequal,  and    (13) 

of  unequal 
phase  angles. 


THREE-PHASE  SYSTEM 


461 


EY 


Ii   — 


1+2YZ 

EY 

1+2YZ 
EY 


.2  1+2FZ 

/3      =    0 


C.    Two  Circuits  Loaded,  One  Unloaded 

Yl  =  F2  =  F,  F3  =  0, 

£i\    =    ^2    —    ^3    =    u • 


Substituting  this  in  equations  (6),  it  is 
e  E  -  Efi(l  +  2  FZ)  +     '2FZ  =  01 


2  FZ)  +  E'x 


O 


loaded  branches. 


-  J0'8  + 


#'2)FZ  =  0     unloaded  branch. 


or,  since 


(13) 


(13) 


TJT  TTIf  77T/     ~\7  rj     f\ 

Ju    —    zv   3    —    £!/3lZ/    —    U, 

E 


YZ' 


thus, 


I  +  4  YZ  +  3  F2Z2 
1  +  4  FZ  +  3  F2Z2 


E's  = 


1  +  YZ 


loaded  branches. 


unloaded  branch. 


(14) 


As  seen,  with  unsymmetrical  distribution  of  load,  all  three 
branches  become  more  or  less  unequal,  and  the  phase  displace- 
ment between  them  unequal  also. 


CHAPTER  XXXVII 
QUARTER-PHASE  SYSTEM 

310.  In  a  three- wire  quarter-phase  system,  or  quarter-phase 
system  with  common  return-wire  of  both  phases,  let  the  two 
outside  terminals  and  wires  be  denoted  by  1  and  2,  the  middle  - 
wire  or  common  return  by  0. 

It  is  then, 

EI  —  E  =  e.m.f.  between  0  and  1  in  the  generator. 
Ez  =  JE  =  e.m.f.  between  0  and  2  in  the  generator. 
Let  1 1  and  1 2  =  currents  in  1  and  in  2, 
I0  =  current  in  0, 

Zj  and  Z2  =  impedances  of  lines  1  and  2, 
Z0  =  impedance  of  line  0, 

FI  and  F2  =  admittances  of  circuits  0  to  1,  and  0  to  2, 
I' i  and  I' 2  =  currents  in  circuits  0  to  1,  and  0  to  2, 
E'i  and  E'%  =  potential  differences  at  circuit  0  to  1,  and 

0  to'2. 
it  is  then,  /i  +  72  +  70  =  0, 


T  /  T          I        7    \  .       I  \*-) 

or,  IQ  =  - 

that  is,  IQ  is  common  return  of  /i  and  72. 

Further,  we  have: 

E',  =    E  -  /iZi  +  /oZ0  =    E  -  7,(Zi  +  Z0)  -  /2Z0    , 
E'2  =  JE  -  72Z0  +  /oZ0  =  JE  -  /2(Z2  +  Z0)  -  V  " 

and  7i  =  F^'i 

7     _  v  T?'  ^Q^ 

f  2  —  1 1&  2  (o; 

Substituting  (3)  in  (2),  and  expanding, 


'i  =  E 


.(l  +  FiZo  +  FiZi)  (1  +  K2Z0  +  F2Z2)  - 


+  FiZo  +  FiZO  (1  +  F2Z0  +  F2Z2)  - 
462 


(4) 


QUARTER-PHASE  SYSTEM  463 

Hence,  the  two  e.m.fs.  at  the  end  of  the  line  are  unequal  in 
magnitude,  and  not  in  quadrature  any  more. 
311.  SPECIAL  CASES: 

A.    Balanced  System 

Zo=vi; 

Fi  =  F2  =  F. 
Substituting  these  values  in  (4),  gives: 


E 

I     -4-  X  A.  I  A.    V  '/.    _J_   V  ZL  I  ZL     V  * '/.  '*• 

(5) 


1  +  V§  (1  +  A/2)  YZ  +  (1  +  \/2)  F2Z2 

1  + (1.707  + 0.707  j)FZ 
1  +  3.414  YZ  +  2.414  F2Z2 


h  A/2)  FZ  +  (1  +  -v/2)  F2Z 
1  +  (1.707  + 0.707  y)  FZ 
"  J    1  +  3.414  FZ  +  2.414  F2Z2 

Hence,  the  balanced  quarter-phase  system  with  common  re- 
turn is  unbalanced  with  regard  to  voltage  and  phase  relation, 
or  in  other  words,  even  if  in  a  quarter-phase  system  with  common 
return  both  branches  or  phases  are  loaded  equally,  with  a  load 
of  the  same  phase  displacement,  nevertheless  the  system  becomes 
unbalanced,  and  the  two  e.m.fs.  at  the  end  of  the  line  are  neither 
equal  in  magnitude,  nor  in  quadrature  with  each  other. 

B.    One  Branch  Loaded,  One  Unloaded 

Zi  =  Z2  =  Z, 
Z 


(a)  F!  =  0,   F2  =  F, 

(b)  Yl  =  F,   F2  =  0. 


464          ALTERNATING-CURRENT  PHENOMENA 
Substituting  these  values  in  (4),  gives: 


(a) 


(6) 


1  +  YZ 
E 


V2 


i  + 


—    7?  I   1 
=  &      I  — 


2.414  + 


FZ 


V2 
1 


E 


1  +  1.707  YZ 

1 


V2 


=  E 


V2 
1 


1  +  1.707  YZ 


E\  = 


V2 


1  + 
=JE\ 


JE  \  1  + 


2.414  +• 


1.414 
~YW 


(6) 


(7) 


These  two  e.m.fs.  are  unequal,  and  not  in  quadrature  with  each 
other. 

But  the  values  in  case  (a)  are  different  from  the  values  in  case 
(6). 

That  means: 

The  two  phases  of  a  three-wire,  quarter-phase  system  are 


QUARTER-PHASE  SYSTEM  465 

unsymmetrical,  and  the  leading  phase,  1,  reacts  upon  the  lagging 
phase,  2,  in  a  different  manner  than  2  reacts  upon  1. 

It  is  thus  undesirable  to  use  a  three-wire,  quarter-phase  system, 
except  in  cases  where  the  line  impedances,  Z,  are  negligible. 

In  all  other  cases,  the  four-wire,  quarter-phase  system  is  pref- 
erable, which  essentially  consists  of  two  independent  single-phase 
circuits,  and  is  treated  as  such. 

Obviously,  even  in  such  an  independent  quarter-phase  system, 
at  unequal  distribution  of  load,  unbalancing  effects  may  take 
place. 

If  one  of  the  branches  or  phases  is  loaded  differently  from  the 
other,  the  drop  of  voltage  and  the  shift  of  the  phase  will  be  differ- 
ent from  that  in  the  other  branch;  and  thus  the  e.m.fs.  at  the  end 
of  the  lines  will  be  neither  equal  in  magnitude,  nor  in  quadrature 
with  each  other. 

With  both  branches,  however,  loaded  equally,  the  system 
remains  balanced  in  voltage  and  phase,  just  like  the  three-phase 
system  under  the  same  conditions. 

Thus  the  four-wire,  quarter-phase  system  and  the  three-phase 
system  are  balanced  with  regard  to  voltage  and  phase  at  equal 
distribution  of  load,  but  are  liable  to  become  unbalanced  at 
unequal  distribution  of  load;  the  three-wire,  quarter-phase 
system  is  unbalanced  in  voltage  and  phase,  even  at  equal  dis- 
tribution of  load. 


30 


APPENDIX 


ALGEBRA  OF  COMPLEX  IMAGINARY  QUANTITIES 
("See  Engineering  Mathematics") 

INTRODUCTION 

312.  The  system  of  numbers,  of  which  the  science  of  algebra 
treats,  finds  its  ultimate  origin  in  experience.     Directly  derived 
from  experience,  however,  are  only  the  absolute  integral  numbers; 
fractions,  for  instance,  are  not  directly  derived  from  experience, 
but    are    abstractions    expressing    relations    between    different 
classes  of  quantities.     Thus,  for  instance,  if  a  quantity  is  divided 
in  two  parts,  from  one  quantity  two  quantities  are  derived,  and 
denoting  these  latter  as  halves  expresses  a  relation,  namely,  that 
two  of  the  new  kinds  of  quantities  are  derived  from,  or  can  be 
combined  to  one  of  the  old  quantities. 

313.  Directly   derived   from   experience   is   the   operation   of 
counting  or  of  numeration, 

a,  a  +  1,  a  +  2,  a  -f-  3   .    .    .    . 
Counting  by  a  given  number  of  integers, 

1  +  1  +  1    .    .    .    +•  1 

a  H =  c, 

o  integers 

introduces  the  operation  of  addition,  as  multiple  counting, 

a  +  b  =  c. 
It  is 

a  +  b  =  b  +  a; 

that  is,  the  terms  of  addition,  or  addenda,  are  interchangeable. 
Multiple  addition  of  the  same  terms, 

a  +  a  +  a  +    .    .    .    +a 

b  equal  numbers 
introduces  the  operation  of  multiplication, 

a  X  b  =  c. 
466 


APPENDIX  467 

It  is 

a  X  b  =  b  X  a, 

that   is,    the    terms    of    multiplication,    or    factors,    are   inter- 
changeable. 

Multiple  multiplication  of  the  same  factors, 

a  X  a  X  a  X  X  a 


b  equal  numbers 
introduces  the  operation  of  involution, 

ab  =  c. 
Since 

ab  is  not  equal  to  &°, 

the  terms  of  involution  are  not  interchangeable. 

314.  The  reverse  operation  of  addition  introduces  the  opera- 
tion of  subtraction. 
If 

a  +  b  =  c, 
it  is 

c  —  b  =  a. 

This  operation  cannot  be  carried  out  in  the  system  of  absolute 
numbers,  if 

b  >  c. 

Thus,  to  make  it  possible  to  carry  out  the  operation  of  sub- 
traction under  any  circumstances,  the  system  of  absolute  num- 
bers has  to  be  expanded  by  the  introduction  of  the  negative 
number, 

-  a  =  (-  1)  X  a, 
where 

(—  1)  is  the  negative  unit. 

Thereby  the  system  of  numbers  is  subdivided  in  the  positive 
and  negative  numbers,  and  the  operation  of  subtraction  possible 
for  all  values  of  subtrahend  and  minuend.  From  the  definition 
of  addition  as  multiple  numeration,  and  subtraction  as  its  inverse 
operation,  it  follows: 

c  -  (-  b)  =  c  +  6, 
thus: 

(-  1)  X  (-  1)  =  1; 

that  is,  the  negative  unit  is  defined  by  (—  I)2  =  1. 


468         ALTERNATING-CURRENT  PHENOMENA 

315.  The  reverse  operatiou  of  multiplication  introduces  the 
operation  of  division. 
If 

a  X  b  =  c, 
it  is 

c 

h  ~  °" 

In  the  system  of  integral  numbers  this  operation  can  only  be 
carried  out  if  b  is  a  factor  of  c. 

To  make  it  possible  to  carry  out  the  operation  of  division 
under  any  circumstances,  the  system  of  integral  numbers  has  to 
be  expanded  by  the  introduction  of  the  fraction, 

|  i     i  i*f> 

where  T  is  the  integer  fraction,  and  is  denned  by 


316.  The  reverse  operation  of  involution  introduces  two  new 
operations,  since  in  the  involution, 

ab  =  c, 

the  quantities  a  and  b  are  not  reversible. 
Thus 

\/c  =  a,  the  evolution, 
loga  c  =  b,  the  logarithmation. 

The  operation  of  evolution  of  terms,  c,  which  are  not  complete 
powers,  makes  a  further  expansion  of  the  system  of  numbers 
necessary,  by  the  introduction  of  the  irrational  number  (endless 
decimal  fraction),  as  for  instance, 

\/2  =  1.414213.    .    . 

317.  The  operation  of  evolution  of  negative  quantities,  c,  with 
even  exponents,  b,  as  for  instance, 

2  , 


makes  a  further  expansion  of  the  system  of  numbers  necessary, 
by  the  introduction  of  the  imaginary  unit 


APPENDIX  '     469 

Thus 

2  /- 2  / 2  /- 

where:  V=  o_  ==  V=  1  X  Vi 

\/^  1  is  denoted  by  j. 

Thus,  the  imaginary  unit,  j,  is  defined  by 

By  addition  and  subtraction  of  real  and  imaginary  units,  com- 
pound numbers  are  derived  of  the  form, 

a  +  jb, 

which   are  denoted  as  complex  imaginary  numbers,   or  general 
numbers. 

No  further  system  of  numbers  is  introduced  by  the  operation 
of  evolution. 

The  operation  of  logarithmation  introduces  the  irrational  and 
imaginary  and  complex  imaginary  numbers  also,  but  no  further 
system  of  numbers. 

318.  Thus,  starting  from  the  absolute  integral  numbers  of 
experience,  by  the  two  conditions: 

1st.  Possibility  of  carrying  out  the  algebraic  operations  and 
their  reverse  operations  under  all  conditions, 

2d.  Permanence  of  the  laws  of  calculation, 

the  expansion  of  the  system  of  numbers  has  become  necessary, 
into 

positive  and  negative  numbers, 

integral  numbers  and  fractions, 

rational  and  irrational  numbers, 

real  and  imaginary  numbers  and  complex  imaginary  numbers. 

Therewith  closes  the  field  of  algebra,  and  all  the  algebraic 
operations  and  their  reverse  operations  can  be  carried  out  ir- 
respective of  the  values  of  terms  entering  the  operation. 

Thus  within  the  range  of  algebra  no  further  extension  of  the 
system  of  numbers  is  necessary  or  possible,  and  the  most  general 
number  is 

a  +  jb, 

where  a  and  b  can  be  integers  or  fractions,  positive  or  negative, 
rational  or  irrational. 

Any  attempt  to  extend  the  system  of  numbers  beyond  the 
complex  quantity,  leads  to  numbers,  in  which  the  factors  of  a 
product  are  not  interchangeable^  in  which  one  factor  of  a  product 


470         ALTERNATING-CURRENT  PHENOMENA 

may  be  zero  without  the  product  being  zero,  etc.,  and  which  thus 
cannot  be  treated  by  the  usual  methods  of  algebra,  that  is,  are 
extra-algebraic  numbers.  Such  for  instance  are  the  double  fre- 
quency vector  products  of  Chapter  XV. 

ALGEBRAIC  OPERATIONS  WITH  GENERAL  NUMBERS 
319.  Definition  of  imaginary  unit: 

J2  =  -  1. 

Complex  imaginary  number: 

A  =  a  +  jb. 
Substituting : 

a  =  r  cos  0, 
6  =  r  sin  0, 
it  is 

A  =  r(cos  0  +  j  sin  0), 

where 


r  =  vector, 

0  =  amplitude  of  general  number,  A. 


Substituting : 

COS0 

sin  0 

it  is 


where  e  =    im  (l  +  £)"  =  .2 


A  =  r<P, 


j  x  2  x  3  x    .    .         < 

is  the  basis  of  the  natural  logarithms. 
Conjugate  numbers  are  called: 

a  H-  jb  =  r(cos  ft  +  j  sin  0)  =  re^, 
and  a  -  jb  =  r(cos  [-  0]+  j  sin  [-  0])  =  r(cqs  0  -  j  sin  0)  =  re~3f*, 

it  is 

(a  -h  jb)(a  -  jb)  =  a2  +  b2  =  r2. 


APPENDIX  471 

Associate  numbers  are  called: 

a  +  jb  =  r  (cos  0  +  j  sin  0)  =  re^, 
and 

6  +  ja  =  r  (cos  [|  -  /s]  +  j  sin  [|  -  0])   =  ni(*  *  ')> 

it  is 

(a  +  jb)(b  +  ja)  =  j(a2  +  62)  =  jr2. 
If 

a  -f-  J6  =  a'  +  jb', 
it  is 

a  =  a', 


6  =  6'. 

If 

a-hj6  =  0; 

it  is 

a  =  0, 

6=0. 

320.  Addition  and  Subtraction: 

(a  +  jb)  ±  (a'  +  jb')  =  (a  +  a')  +  j  (6  +  60- 
Multiplication: 

(a  +  J6)(a'  +  jb')  =  (aa'  -  66')  +  j  (a&'  +  ba')', 

or       r  (cos  |8  +  j  sin  0)  X  r'(cos  £'  +  j  sin  0')  =  rr'  (cos  [0  +  0'] 

+  j  sin  [0  +  0']); 
or         re^XrV^'  =  rrV^  +  ^ 

Division: 

Expansion  of  complex  imaginary  fraction,  for  rationalization 
of  denominator  or  numerator,  by  multiplication  with  the  con- 
jugate quantity: 

a+jb         (a+jbf)  (af  -JV)  _  (aa'  +  W)  +  j  (baf  -  abf) 
')  (a'  -  J6')  =  a'2  +  6'2 


(a'  +  J6')  (a  -  jb)       (aaf  +  66')  +  j  (abf  -  ba')  ' 
or, 


or 


472         ALTERNATING-CURRENT  PHENOMENA 

involution: 

(a+jb)n=  {r(cos/3  +  jsin/3))n   =  {re^}n 
=  rn(cos  wj8  +  j  sin  n/3)  =  rV71^; 


W  y-    /  /3      .         .       .        /3\  n   y  -  j 

:  v  r  (cos  -  +  j  sm  -)   =  v  re  n. 
321.  .Roo^  o/  the  Unit: 

vT  =  +  1,  --  i; 

u-1      -  !  +  J  V3      -  1  -JV3. 
=     +1,-      ~2-  -2~ 

=  +1,  -  1,  +j,  -  j; 

6/T       j.  1    ±l±r\/3  -l+j\/3  -1-JV3    +1-JA/3 

VI  =+1,  -    -y-  ^-~>-1'"-2-  ^""J 

8/T-          LI  •          .   +1+J    +1-J    -1+.7    -1-j. 

-- 


n/-  27T/C     ,       .     .       27T/C  2^ 

V  1  =  cos  -  —  h  J  sm  --  =  e  »  ,  fc  =  0,  1,  2   .    .    .    .   n  —  1. 

n  n 

322.  Rotation: 

In  the  complex  imaginary  plane,  multiplication  with 

n/-  2-JT     ,  27T  ^ 

V  1   =  cos  ---  \-  j  sin  -  -  =  e  n 
n  n 

means  rotation,  in  positive  direction,  by  —  of  a  revolution, 

multiplication  with  (—1)  means  reversal,  or  rotation  by  180°, 
multiplication  with  (+  j)  means  positive  rotation  by  90°, 
multiplication  with  (  —  j)  means  negative  rotation  by  90°. 

323.  Complex  Imaginary  Plane: 

While  the  positive  and  negative  numbers  can  be  represented 
by  the  points  of  a  line,  the  complex  imaginary  numbers  or  general 
numbers  are  represented  by  the  points  of  a  plane,  with  the  hori- 
zontal axis,  A'OA,  as  real  axis,  the  vertical  axis,  B'OB,  'as  im- 
aginary axis.  Thus  all 

the  positive  real  numbers  are  represented  by  the  points  t)f  half- 

axis  OA  toward  the  right; 
the  negative  real  numbers  are  represented  by  the  points  of  half- 

axis  OA'  toward  the  left; 


APPENDIX  473 

the  positive  imaginary  numbers  are  represented  by  the  points  of 

half-axis  OB  upward; 
the  negative  imaginary  numbers  are  represented  by  the  points  of 

half-axis  OB'  downward; 
the  complex  imaginary  or  general  numbers  are  represented  by  the 

points  outside  of  the  coordinate  axes. 


INDEX 


Absolute  values  of  complex  quanti- 
ties, 37 
Actual  generated  e.m.f.,  alternator, 

272 
Admittance,  55 

of  dielectric,  154 

due  to  eddy  currents,  137 

to  hysteresis,  129 

Admittivity  of  dielectric  circuit,  160 
Air-gap  in  magnetic  circuit,  119,  132 
Ambiguity  of  vectors,  39 
Amplitude,  6,  20 
Apparent  capacity  of  distorted  wave, 

386 
efficiency    of   induction   motor, 

234 

impedance  of  transformer,  201 
torque    efficiency   of   induction 

motor,  234 
Arc  causing  harmonics,  353 

as  pulsating  resistance,  352 
volt-ampere  characteristic,  354 
wave  construction,  355 
Armature  reaction  of  alternator,  260, 

272 
Average  value  of  wave,  11 

Balanced  polyphase  system,  397 
Balance  factor  of  polyphase  system, 

406 
Brush  discharge,  112 

Cable,  topographical  characteristic, 

42 
Capacity,  4,  9 

of  line,  174 
Choking  coil,  96 

Circuit    characteristic    of    line    and 
cable,  44 

dielectric  and  dynamic,  159 

factor  of  general  wave,  383 
Coefficient  of  eddy  currents,  138 

of  hysteresis,  123 
Combination  of  sine  waves,  31 


Compensation  for  lagging  currents 

by  condensance,  72 
Condensance  in  symbolic  expression, 

36 
Condenser  as  reactance  and  suscep- 

tance,  96 

with  distorted  wave,  384 
motor  on  distorted  wave,  392 
motor,    single-phase   induction, 

249,  257 

synchronous,  339 

Conductance  of  circuit  with  induc- 
tive line,  84 
direct  current,  55 
due  to  eddy  currents,  137 
effective,  111 

due  to  hysteresis,  126 
parallel  and  series  connection, 

54 

Conductivity,  dielectric,  153 
of  dielectric  circuit,  160 
Constant  current  from  constant  po- 
tential, 76 

synchronous  motor,  337 
potential  constant  current  trans- 
formation, 76 

Consumed  voltage,  by  resistance,  re- 
actance, impedance,  23 
Control  of  voltage  by  shunted  sus- 

ceptance,  89 
Corona,  112,  161 

of  line,  174 

Counter  e.m.f.  of  impedance,  react- 
ance, resistance,  self-induc- 
tion, 23 

of   synchronous  motor,  24,  315 
Crank  diagram,  19 

and  polar  diagram,  comparison, 

51 

Critical  voltage  of  corona,  166 
Cross  currents  in  alternators,  293 
Cross  flux,  magnetic  of  transformer, 

187 
Cycle,  magnetic  or  hysteresis,  114 


475 


476 


INDEX 


Delta  connection  of  three-phase  sys- 
tem, 416 
current  in  three-phase  system, 

417 

delta  transformation,  425 
Y  transformation,  425 
voltage  in  three-phase  system, 

417 

Demagnetizing  effect  of  eddy  cur- 
rents, 142 

Diametrical    connection    of    trans- 
formers, six-phase,  429 
Dielectric  circuit,  159 
density,  152 
field,  150 

hysteresis,  112,  150 
strength,  161 
Direct-current     system,     efficiency, 

441 

Displacement  current,  152 
Disruptive  gradient,  165 
Distortion  by  magnetic  field,  resist- 
ance and  reactance  pulsa- 
tion, 342 

of  magnetizing  current,  117 
of  wave,  see  Harmonics 

by  hysteresis,  116 
Distributed  capacity,  168 
Double  delta  connections  of  trans- 
formers to  six-phase,  428 
frequency    power    and    torque 

with  distorted  wave,  381 
quantities,  180 
peak  wave,  370 
T  connections  of  transformers 

to  six-phase,  430 
Y  connection  of  transformers  to 

six-phase,  429 
Drop  of  voltage  in  line,  25 
Dynamic  circuit,  159 

Eddy  currents,  112 

admittance,  137 

coefficient,  138 

conductance,  137 

in  conductor,  144 

loss  with   distorted  wave,   377 

of  power,  136 
Effective  circuit  constants,  168 


Effective  circuit  conductance,  111 
power,  180 
reactance,  112 
resistance,  2,  5,  9,  111 
susceptance,  112 
value  of  wave;  11 

in  polar  diagram,  53 
Efficiency  of  circuit  with  inductive 

line,  88,  95 
induction  motor,  234 
Electrostatic,  see  Dielectric 
E.m.f.  of  self-induction,  123 
Energy  distance  of  dielectric  field, 

165 

flow  in  polyphase  system,  406 
and    torque    as    component    of 
double    frequency    vector, 
186 

Epoch,  6 
Equivalent   circuit   of  transformer, 

202 

sine  wave  in  polar  diagram,  53 
single-phase  circuit  of  polyphase 

system,  448 
Excitation  of  induction  generator, 

238 

Exciter  of  induction  generator,  238 
Exciting    admittance    of    induction 

motor,  211 

current  of  induction  motor,  211 
single-phase  induction  motor, 

247 
transformer,  189 

Field  characteristic  of  alternator,  265 
Fifth  harmonic,  370 ' 
Five-wire  system,  efficiency,  466 
Flat  top  wave,  370 

zero  wave,  370 
Foucault  currents,  113 
Four-phase  system,  397 

wire  systems,  efficiency,  466 
Frequency,  6 

General  wave,  symbolism,  379 
Generator,  induction,  237 

Harmonics,  7 

caused  by  arc,  353 


INDEX 


477 


Harmonics  of  current,  341 

by  hysteresis,  116,  358 

by  three-phase  transformer,  363 

of  voltage,  341 
Hedgehog  transformer,  189 
Hemisymmetrical     polyphase     sys- 
tem, 404 

Higher  harmonics,  see  Harmonics 
Hysteresis,  admittance,  129 

advance  of  phase,  122,  130 

coefficient,  123 

conductance,  126 

cycle,  115 

unsymmetrical,  13.5 

dielectric,  150 

dielectric  and  magnetic,  112 

in  line,  174 

loss,  122 

with  distorted  wave,  377 

power  current,  117 
voltage,  123 

Imaginary  power,  186 
Impedance,  2,  9 

apparent,  of  transformer,  201 
of  induction  motor,  211 
in  series  with  circuit,  69 
series  and  parallel  connections, 

55,  59 

in  symbolic  expression,  35 
synchronous,  of  alternator,  277 
Independent  polyphase  system,  397 
Inductance,  3,  9 

factor  of  general  wave,  382 
Induction  generator,  237 

machine  as  inductive  reactance, 

96 
motor,  208 

on  distorted  wave,  392 
Inductive   devices,    starting   single- 
phase  induction  motor,  246 
line,  maximum  power,  82 
Inductor  alternator,  unsymmetrical 

magnetic  cycle,  135 
Influence,    electrostatic,    from   line, 

174 

Instantaneous  value,  11 
Intensity  of  wave,  20 
Interlinked  polyphase  system,  397 


Inverted    three-phase    system,  398, 

408,  413 
efficiency,  466 
Ironclad  circuit,  119,  131 

wave  shape  distortion,  358,  361 
Iron  wire  and  eddy  currents,  140 
unequal    current    distribution, 
147 

j  as  distinguishing  index,  32 

as  imaginary  unit,  33 
Joule's  law,  1,  5 

Kirchhoff's  laws,  direct  current,  1 
in  crank  diagram,  22,  60 
in  polar  diagram,  49 
in  symbolic  expression,  34 

Lag    in    alternator,    demagnetizing, 

260 

of  current,  21 

in  synchronous  motor,  magnet- 
izing, 261 
Laminated  iron  and  eddy  currents, 

138 

Lead  in  alternator,  magnetizing,  260 
of  current,  21 

by  synchronous  condenser,  339 
in  synchronous  motor,  demag- 
netizing, 261 
Leakage,  112,  151  -< 

currents  through  dielectric,  152 

in  transformer,  189 
of  line,  174 

reactance  of  transformer,  187 
Line  capacity,  169 
phase  control,  99 
power  factor  control,  99 
topographic  characteristic,  43 
Load  curves  of  synchronous  motor, 
333 

Magnetic  cycle,  114 
hysteresis,  112 
Magnetizing  current,  117 
Maximum  output  of  inductive  line, 

83 

non-inductive  circuit  and  in- 
ductive line,  81 


478 


INDEX 


Maximum  power  of  induction  mo- 
tor, 222 

torque  of  induction  motor,  219 
Mean  value  of  wave,  12 
Metering  of  polyphase  systems,  442 
M.m.f.,  rotating,  of  polyphase  sys- 
tem, 401 

Molecular  friction,  112 
Monocyclic     connection     of     trans- 
formers, 428 
devices,     starting    single-phase 

induction  motor,  246 
system,  409 

Multiple  phase  control,  108 
Mutual  inductance,  174 
induction,  147 
inductive  reactance  of  line,  174 

Neutral  voltage  of  three-phase  trans- 
former, 367 

Nominal  generated  e.m.f.  of  alter- 
nator, 263,  276,  282 

Non-inductive  circuit  and  inductive 
line,  79,  81 

Ohm's  law,  1 

Open  delta  transformation,  427 
Oscillating  waves,  175 
Output,  see  Power 

of   circuit   with   inductive  line, 
82,  95 

in  phase  control,  104 

Parallel  connection  of  admittances, 

59 

of   resistances   and   conduct- 
ances, 54 

operation  of  alternators,  292 
Parallelogram  of  sine  waves,  22 

in  polar  diagram,  48 
Peaked  waves,  370 
Peaks  of  voltage  by  wave  distortion, 

360,  367 
Permittivity,  152 

of  dielectric  circuit,  160 
Phase,  6,  20 

advance    angle    by    hysteresis, 
122,  130 


Phase  characteristic  of  synchronous 

motor,  328 
control,  97 

multiple,  108 

difference  in  transformer,  29 
splitting  devices  starting  single- 
phase  induction  motor,  246 
Polar    coordinates     of     alternating 

waves,  46 
and  crank  diagram,  comparison, 

51 
Polarization,  4 

cell  as  condensive  reactance,  96 
Polycyclic  systems,  409 
Polygone  of  sine  waves,  22 

in  polar  diagram,  48 
Polyphase    and    constituent    single- 
phase  circuit,  448 
Power,  see  Output 

characteristics  of  polyphase  sys- 
tems, 409 

components  of  current  and  volt- 
age, 168 

consumption  by  corona,  165 
as  double  frequency  vector,  180 
factor  of  arc,  356 

correction     by     synchronous 

condenser,  339 
of  dielectric  circuit,  152 
of  general  wave,  382 
of  induction  motor,  234 
phase  control,  99 
of  general  wave,  381 
of  induction  motor,  216,  222 
loss  in  dielectric,  157 
of  sine  wave,  22 
vector  denotation,  179 
of  wave  in  polar  diagram,  49 
Primary  admittance  of  transformer, 

197 

impedance  of  transformer,  198 
Pulsating  magnetic  circuit,  135 

wave,  11 

Pulsation  of  magnetic  circuit,  react- 
ance and  resistance,  342 

Quadrature  components  of  alterna- 
tor armature  reaction  and 
reactance,  282 


INDEX 


479 


Quadrature  flux  of  single-phase  in- 
duction motor,  245 
Quarter-phase  system,  398 

efficiency,  466 

three-phase  transformation,  423 
Quintuple  harmonic,  see  Fifth  har- 
monic 

Radiation  from  line,  174 
Ratio  of  transformer,  197 
Reactance,  2,  9 
effective,  112 
in  phase  control,  103 
in  series  with  circuit,  63 
in  symbolic  expression,  35 
synchronous,  of  alternator,  277 
Reactive  component  of  current  and 

voltage,  168 
power,  180 

with  general  wave,  382 
Rectangular  components,  31 
Reduction  of  polyphase  system  to 

single-phase  circuit,   448 
Regulation  of  circuit  with  inductive 

line,  82,  86 

curve  of  alternator,  290 
Resistance,  effective,  2,  5,  9,  111 
of  line,  174 
parallel  and  series  connection, 

54 

in  series  with  circuit,  60 
in    starting    induction    motor, 

224 

in  symbolic  expression,  35 
Resolution  of  sine  waves,  31 
Resonance   of   condenser  with   dis- 
torted wave,  387 
by  harmonics,  373 
Ring  connection  of  polyphase  sys- 
tem, 416 
current    in    polyphase    system, 

417 
voltage    in    polyphase    system, 

417 
Rise  of  voltage  of  circuit  by  shunted 

susceptance,  94 

Rotating  field  of  symmetrical  poly- 
phase system,  401 
Ruhmkorff  coil,  7 


Saturation,  magnetic,  induction  gen- 
erator, 238 
Saw-tooth  wave,  370 
Screening  effect  of  eddy  currents,  142 
Secondary     impedance     of     trans- 
former, 198 

Self-excitation  of  induction  genera- 
tor, 238 

Self -inductance,  174 
Self-inductive  reactance  of  alterna- 
tor, 261 

of  transformer,  187 
voltage,  123 
Series  connection  of  impedances,  55, 

59 

of    resistances   and    conduct- 
ances, 54 

impedance  in  circuit,  69 
operation  of  alternators,  294 
reactance  in  circuit,  63 
resistance  in  circuit,  60 
Sharp  zero  wave,  370 
Short  circuit  of  alternator,  273,  288 
Shunted    condensance    and    lagging 

current,  72 

Silent  discharge  from  line,  174 
Single-phase     cable,     topographical 

characteristic,  42 
circuit  equivalent  to  polyphase 

system,  448 
efficiency,  466 
induction  motor,  245 
system,  398 

Slip  of  induction  motor,  208 
Spheres,  dielectric  field,  164 
Stability  of  induction  motor,  238 
Star  connection  of  polyphase  system, 

415 
current    in    polyphase    system, 

417 
voltage    in    polyphase    system, 

417 

Starting  devices  of  single-phase  in- 
duction motor,  245 
torque  of  induction  motor,  223 
single-phase  induction  motor, 

252 
Susceptance,  55 

of  circuit  with  inductive  line,  82 


480 


INDEX 


Susceptance,  effective;   112 
Susceptivity,  dielectric,  153,  160 
Symbolic  expression  of  power,  181 
Symmetrical  polyphase  system,  396 
Synchronizing  power  of  alternators, 

294 
Synchronous  condenser,  339 

converter  for  phase  control,  98 
impedance  of  alternator,  277 
machine  as  shunted  susceptance, 

96 
motor,    fundamental    equation, 

316 

for  phase  control,  98 
supplied  by  distorted  wave, 

389 

reactance  of  alternator,  262,  272 
watts  as  torque,  233 

T  connection  of  transformers,   427 
Terminal  voltage  of  alternator,  263 
Tertiary    circuit    with     condenser, 
single-phase  induction  mo- 
tor, 249 
Third  harmonic,  369 

in  three-phase  system,  364 
Three-phase  line,  topographic  char- 
acteristic, 43 
quarter-phase     transformation, 

423 

system,  397 
efficiency,  466 
voltage  drop,  41 
transformer,    wave    distortion, 

363 

Three-wire  single-phase  system,  effi- 
ciency, 466 
Time  constant,  3 

and  crank  diagram,  comparison, 

51 

diagram  of  alternating  wave,  48 
Topographic  characteristic  of  cable 

and  line,  42 
Torque  as  double  frequency  vector, 

185 

efficiency   of   induction   motor, 
234 


Torque  of  induction  motor,  216,  219, 

223 

single-phase    induction    motor, 
248,  252 

Transformation  by  two  transform- 
ers, of  polyphase  systems, 
422 

Transformer,  187 
diagram,  26,  30 
equivalent  circuit,  202 

Transmission  line,  see  Line 

Treble  peak  wave,  370 

Triple  harmonic,  see  Third  harmonic 

True  power  of  generator  wave,  sym- 
bolic, 382 

Unbalanced  polyphase  system,  397 
quarter-phase  system,  463 
three-phase  system,  461 

Unequal  current  distribution  in  con- 
ductor, 144 

Unsymmetrical  hysteresis  cycle,  135 
polyphase  system,  396 

V    connection    of    transformers    on 

three-phase  system,  427 
Vector  power,  179 
Virtual    generated    e.m.f.  of    alter- 
nator, 272 
Voltage    of    circuit    with  inductive 

line,  82,  86 
control  by  shunted  susceptance, 

89 
by    synchronous    condenser, 

339 
peaks  by  wave  distortion,  360, 

367 
phase  control,  99 

Y  connection  of  three-phase  system, 

416 
current  in  three-phase  system, 

417 

Delta  transformation,  426 
voltage  in  three-phase  system, 

417 
Y  transformation,  426 


GENERAL  LIBRARY 
UNIVERSITY  OF  CALIFORNIA— BERKELEY 

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